Preheat the oven to 350°F.
Place the butter in a light-colored saucepan and melt over high heat. Once the butter begins to bubble, reduce the heat to medium-low.
Stirring every so often to keep the milk solids from burning and sticking to the bottom of the pan, continue to cook the butter until it turns an amber-colored brown and the milk solids have darkened. This will take 5 to 8 minutes. It is done when the brown butter smells nutty and toasty. Pour the brown butter into a heat-safe container. Place the butter in the refrigerator to cool as you continue with the recipe.
In a mixing bowl, stir together the flour, baking powder, and salt. Set aside.
Using a stand or hand mixer, beat the eggs and sugar together until they double in volume and are pale in color. When done, the batter should flow like thick ribbons. This will take about 5 minutes.
Add the brown butter, oil and vanilla and mix to combine.
Add half of the dry ingredients and mix on low speed until combined.
Pour in the milk and sour cream. Mix until smooth.
Add in the remaining dry ingredients and mix until just incorporated. Fold the batter together with a large spatula until smooth.
Line two muffin tins with cupcake liners. Use a spring-loaded cookie scoop to evenly distribute the batter. Fill the liners about ⅔ of the way full.
Bake the cupcakes for 20 to 22 minutes, or until a toothpick inserted into their centers comes out clean or with a few moist crumbs. Allow the cupcakes to cool on a wire rack before topping with frosting.