In a bowl, with an electric mixer on medium speed, beat browned butter and sugar until well blended. Beat in egg white and vanilla until smooth, scraping down sides of bowl as needed.
In another bowl, mix flour and baking soda. Stir or beat into butter mixture until well incorporated. Stir in crushed peppermints.
Drop dough in 1/2-teaspoon portions, 1 inch apart, onto buttered or cooking parchment-lined 12- by 15-inch baking sheets.
Bake in a 350 oven until tops of cookies are deep golden and slightly cracked, 8 to 10 minutes; if baking more than one pan at a time, switch pan positions halfway through baking.
With a wide spatula, transfer cookies to racks to cool. If hot cookies start to break, slide a thin spatula under them to release; let stand on pan to firm up, 2 to 5 minutes, then transfer to racks to cool completely.
