Preheat oven to 325F. Line standard muffin tins with paper liners. Whisk together the flour, baking powder and salt.

With an electric mixer on medium-high speed, cream butter and brown sugar until pale fluffy. Add eggs, one at a time, beating until each is incorporated, scraping down sides of bowl as needed. Add flour mixture in three batches, alternating with two additions of buttermilk, and beating until combined after each.

Divide batter evenly among lined cups, filling each three-quarters full. Bake, rotating tins halfway through, until golden brown and a cake tester inserted in centers comes out clean, about 25 minutes. Transfer tins to wire racks to cool 10 minutes; turn out cupcakes onto racks and let cool completely. Cupcakes can be stored up to 3 days at room temperature, or frozen up to 2 months, in airtight containers.

To finish, place cupcakes on a wire rack set over a baking sheet; spoon icing over cupcakes and let set. Cupcakes are best eaten the day they are glazed; keep at room temperature until ready to serve.

Melt butter in a small saucepan 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.