Preheat oven to 325 degrees. In a saucepan, melt the butter over medium heat. Continue to cook, stirring occasionally, until the butter turns a golden brown color. Drain the butter into a small bowl (leaving the foam on top behind) to let cool.
Whisk together flour, baking powder, cinnamon, salt, cloves, and nutmeg. In another bowl, whisk the sugars, eggs, butter, and pumpkin puree. Whisk in the flour mixture until combined.
Divide the batter evenly among paper-lined cups, filling each 3/4 full. Bake until a cake tester inserted in the cupcakes comes out clean. This should take about 20 minutes. Transfer cupcakes to wire racks to cool completely before glazing them.
To make the icing, melt the butter in a small saucepan over medium heat until brown in color. This might take about 10 minutes. Remove from heat and pour the butter into a bowl.
Stir in the powdered sugar, vanilla, and 2 tablespoons of milk to the butter until smooth. You can add more milk by tablespoons if necessary. Use immediately or else it hardens! To finish, spoon about 1 tablespoon of glaze over each cupcake. It's okay if it drips down the sides!