What's the Effect of Browning Butter From a taste perspective, I love the nutty flavor of browned butter.  From a culinary perspective, I'm curious what the process does to the butter.  

I used browned and cooled butter in cookies yesterday.  The cookbook I was working with had another recipe for cookies with melted butter (not browned) and suggested using egg yolks to add some extra fat because melted butter is different than softened butter, but did not explain.  

I followed the tips for adding egg yolks and the cookies came out well, but I'm curious if I could have skipped the egg yolks and cooled the butter until it was the temperature of butter for creaming (which I've read is around 70 degrees F) and creamed it as normal softened butter, or if browning it altered it permanently.
Your recipe is suspect. Melted butter is the same in chemical composition as softened butter. The two react a bit differently in recipes due to the way that they interact with the other ingredients to impart textures: in short, the creamed butter will hold microscopic bubbles that the fully melted butter will not. Adding egg yolks will help the melted butter incorporate, and probably lead to more cakey cookies (compensating for the flatness you get from melted butter), but the effect is hard to isolate.

As for brown butter, what happens is that as moisture evaporates from the butter, the boiling point increases (similar to what happens when making caramel or candy). As the boiling point of the butterfat increases, the temperature of the butter solids (which consist of lactose and milk proteins) begin to caramelize (the sugars) and undergo a maillard reaction (the protein). Both of these reactions will impart unique flavors to the butter, yielding the distinctive brown butter taste.

As the butter heats, the integrity of the fat is damaged. This occurs to all fats, which is why you can only use a batch of fat for deep frying a few times before it needs to be discarded.