How to Make Lobster Bisque
To make lobster bisque, start by steaming the lobster tails and removing the meat from the shells. Next, saute the shells in melted butter along with aromatics like shallots, celery, and garlic. Then, add the tomato paste, diced tomatoes, brandy, stock, and seasonings and simmer the mixture for 30 minutes. Finally, saute the lobster meat in butter, strain the soup broth, and puree the meat with the strained solids before adding it back to the broth.
Cut each tail in half. Carefully cut through the shell of each lobster tail using kitchen shears. Separate the tail in half lengthwise, dividing both the shell and the meat. If you're using frozen lobster tails, you'll need to thaw the tails completely before beginning. You'll likely need to cut through the top shell first, followed by the bottom shell. Cut through the lobster meat after splitting the shell, following the edge of the divided shell. The lobster meat should remain in the shell after you finish cutting the tails in half. If you notice a large vein in the tail, remove that vein before continuing to the next step. Salt and simmer the water. Pour 2 cups (500 ml) of water into a stockpot. Set it on the stove over medium-high heat, then reduce the heat to medium once the water begins to simmer. As soon as the water simmers, add the salt, which should help draw out the flavor of the lobster tails. Steam the tails. Place the tails shell-side down inside the steaming basket, then set the basket over the pot of simmering stock. Cover and cook for 5 to 7 minutes. Steam the lobster tails until they're cooked through. The shells should be bright red, and the meat should feel firm and look opaque. Remove the meat. Use tongs to remove the lobster tails from the steamer basket and set them onto a separate plate. After cooling them for several minutes, dig the meat out of the shells. Set aside the lobster meat until later, keeping it in the refrigerator so that it does not spoil. Do not drain the water. This liquid actually absorbs some of the lobster flavor during the steaming process, so you'll actually benefit from adding it to the soup broth later on. Melt half the butter. Place 2 Tbsp (30 ml) of butter into a large stockpot or Dutch oven. Set it on the stove over medium-high heat and allow the butter to melt. Once the butter melts, carefully turn the pan so that the butter can coat the entire bottom evenly. Note that butter gives the soup a richer flavor, but if desired, you could use olive oil instead. Simply heat the oil until it glistens. Meanwhile, flatten the shells. Place the empty shells on the cutting board, insides-down. Using the heel of the hand or the flat side of a large kitchen knife, press down on the shells until they crack and become mostly flat. By flattening the shells, you guarantee that more of the surface will touch the pan when you saute them in the next step. Saute the shells. Place the shells in the melted butter and cook, stirring frequently, for 3 to 5 minutes. While you can technically use the shells without cooking them in butter, taking the time to perform this extra step draws more flavor out of them, which will also improve the flavor of the finished soup. Add the aromatics. Toss the shallots, celery, and garlic into the stockpot with the lobster. Continue to cook, stirring frequently, for another 2 minutes. Both the garlic and shallots should become notably more fragrant. The garlic should turn a light golden brown, but the shallots should look slightly translucent. Deglaze the pan. Pour the dry white wine into the stockpot. Stir it into the existing ingredients, scraping the bottom of the pan with the spoon as you do. Deglazing the stockpot lifts off any bits of food that have stuck to the bottom of the pan while cooking. These deglazed bits can make the taste of the dish deeper and more complex. Add the remaining base ingredients. Add the tomato paste, undrained diced tomatoes, brandy, stock, and reserved steaming water to the stockpot. Sprinkle in the thyme, paprika, cayenne pepper, and bay leaf. Stir well to combine. The paprika and cayenne pepper give the lobster bisque a slight kick. If you prefer a milder taste, reduce the amount or omit these two ingredients altogether. Simmer for 30 minutes. Bring the soup base to a boil, then reduce the heat to medium-low. Simmer the contents of the stockpot for 30 minutes. Leave the stockpot uncovered during this time. Stir the base occasionally, and adjust the heat as needed to keep the soup at a steady simmer. Melt the remaining butter. Add the remaining 2 Tbsp (30 ml) of butter to a separate skillet. Place the pan on the stove over medium-high heat, and allow the butter to melt and coat the bottom of the pan. Try to time this step so that the lobster meat is ready once the soup base is ready. Typically, you should melt this butter when the base only has roughly 10 minutes left of simmering time. Saute the meat. Cut the reserved lobster meat into large chunks, then toss it into the hot butter. Cook, stirring frequently, for about 3 to 5 minutes or until the meat becomes lightly browned. Separate the meat after you finish cooking it. Reserve four large chunks for the garnish, and prepare the rest for use in the soup itself. Strain the soup. Pour the simmered soup base through a large, fine mesh sieve. Immediately return the liquid to the stockpot. Fish through the solid ingredients and remove the bay leaf. Keep the rest of the solids. Puree the solids and some meat. Toss the remaining soup solids and the soup portion of the lobster meat into a large blender. Puree the mixture until it becomes smooth. While it may seem unusual if you've never done it before, you should include the shells in this puree. Most of the shells will be blended down, and any chunks that remain will be strained out later. Cook the puree with the liquid base. Add the puree back to the soup, mixing well to combine. Simmer the soup for another 10 to 15 minutes. Strain the remaining solids. Line the sieve with cheesecloth, then pour the soup through. Return the liquid to the stockpot and discard the remaining solids. To get as much liquid out as possible, press down on the solids with the back of the mixing spoon, squeezing the liquid out through the cheesecloth and sieve as you do. Add the dairy. Stir the heavy cream and half-and-half into the soup. Bring the soup to a gentle simmer over medium-low heat. Note that it'll be easiest to add the dairy if you previously warm it to room temperature or slightly higher. If you use cold dairy, it may curdle once you add it to the hot soup. Combine the water and starch. In a separate small bowl, whisk together 2 Tbsp (30 ml) of cold water and the tapioca flour or cornstarch. Continue mixing until a smooth slurry forms. The cornstarch slurry acts as a thickener for the soup. Traditional lobster bisque actually uses uncooked rice, and some recipes call for a flour-based roux. Cornstarch slurries are easy to work with, however, and give the finished soup a smooth texture and a shiny appearance. Add the slurry to the soup. Add the cornstarch slurry to the soup, stirring to combine. Cook the soup for another 2 or 3 minutes, or until it thickens. Serve with the remaining lobster meat. Ladle the finished soup into individual serving bowls. Garnish each bowl with a piece of the reserved lobster meat, and enjoy! The soup is best served warm and can be enjoyed immediately, but if you want to increase the flavor even more, refrigerate the soup overnight and reheat it on the stove the following day. 
Cut each tail in half. Salt and simmer the water. Steam the tails. Remove the meat. Melt half the butter. Meanwhile, flatten the shells. Saute the shells. Add the aromatics. Deglaze the pan. Add the remaining base ingredients. Simmer for 30 minutes. Melt the remaining butter. Saute the meat. Strain the soup. Puree the solids and some meat. Cook the puree with the liquid base. Strain the remaining solids. Add the dairy. Combine the water and starch. Add the slurry to the soup. Serve with the remaining lobster meat. 