Brown Butter Icing
½ cup (1 stick) unsalted butter
2 cups sifted confectioners' sugar
2 teaspoons pure vanilla extract
2 tablespoons milk, plus more if needed
Melt butter in a small sauce pan over medium heat, swirling pan occasionally, until nut brown in color, about 10 minutes. Remove from heat, and pour butter into a bowl, leaving any burned sediment behind.
Add confectioners' sugar, vanilla, and 2 tablespoons milk to brown butter; stir until smooth. If necessary, add more milk (up to 2 tablespoons) a little at a time, just until icing is spreadable. Use immediately.