Preheat the oven to 400°F. Put butter in a small saucepan on medium heat, stirring until it turns a dark golden color. Take the pan off the heat. Let the butter solidify again; I do this by placing it in the freezer for about 10 minutes. This allows it to solidify on top and when you stir it, it will be the consistency of softened butter.
When the butter is solid, but still soft, put all the cake ingredients except the milk in a food processor (or mixer) and pulse to a smooth batter. Add the milk down the funnel, pulsing sparingly to form a soft, dropping mixture.
Divide among the paper cups, and cook for 15-20 minutes. Makes 12.
For the frosting
Brown butter the same way as you did before; allow it to come back to room temperature.
Put dulce de leche and browned butter into mixer and cream together. Mix in vanilla, cream, and salt. Mix in powdered sugar 1 cup at a time until desired consistency is reached.
Assembly
Using a cupcake corer or a knife, scoop out the top, center of the cupcake when they're cool and fill with dulce de leche. Frost as desired and sprinkle a little sea salt on top (optional; I like to do this because it cuts the sweetness a bit).