name of the process of heating butter to give it the special aroma One of common toppings for fruit pierogi and several other dishes, usually sweet, is butter that has been melted and heated on a skillet/saucepan until it got a somewhat deeper golden color, and a special unique aroma and flavor.

What is this process called?

(I believe it's also the first phase of preparing clarified butter, but I'm not sure about that.)
It's called browned butter or beurre noisette.  It is essentially heating butter beyond its melting point until the solids start to darken.  E.g.: a recipe is here at Kitchn.