For the Doughnut Muffins

1. Preheat oven to 350 degrees F. Coat a mini-muffin tin with nonstick cooking spray.
2. Beat the butter and sugar for 5 minutes until soft, fluffy and lighter in color. Beat in the eggs, one at a time, until just mixed in.
3. Sift together the flour, baking powder, and salt. Stir a fourth of the dry ingredients into the batter. Then stir in a third of the milk mixture. Continue mixing in the remaining dry and wet ingredients alternately, ending with the dry. Mix until well combined and batter is smooth, but be careful not to overmix.
4. Fill muffin tins with 1.5 tablespoons of batter (about 3/4ths full). Bake at 350 for 14-15 minutes until a toothpick comes out clean.

For the Brown Butter Cinnamon Sugar Crust

1. Melt the butter until it starts to bubble and then foam. When it starts to foam, whisk constantly for 1-2 minutes until it turns brown and you see brown flecks at the bottom of the pan. Immediately remove from heat and pour into a bowl for dipping.
2. Combine the sugar and cinnamon. When the muffins are just cool enough to handle, remove from the tin and dip them on all sides into the melted butter then roll in the cinnamon sugar.

For the Brown Butter Buttercream Frosting

1. Melt the butter until it starts to bubble and then foam. When it starts to foam, whisk constantly for 1-2 minutes until it turns brown and you see brown flecks at the bottom of the pan. Immediately remove from heat and pour into a mixing bowl. Let cool to room temperature.
2. Place the butter, sugar, and vanilla in a large mixing bowl. Beat on medium speed until smooth and creamy, about 3-5 minutes. If necessary, add milk, one tablespoon at a time, beating for 1-2 minutes after each addition until desired consistency is reached.
3. Pipe onto muffins.