I highly recommend making brown butter and heat treated flour the night before so they have time to fully cool! See notes.
Edible Cookie Dough
Using a stand mixer fitted with the paddle attachment, or a large bowl and hand mixer, cream together the room temperature brown butter, brown sugar, granulated sugar, salt, and vanilla until light and fluffy (3-4 minutes).
Add in the flour and milk and mix until just combined. If the dough is too dry, you can add in more milk 1 teaspoon at a time until the dough comes together.
Fold in the chocolate chips and then let chill in the fridge until ready to use.
Cupcakes
Preheat the oven to 350°F and line a 12 cup cupcake pan with cupcake liners.
Whisk the cake flour, baking powder, and salt together in a large bowl.
In a separate bowl, whisk the brown butter, sugar, brown sugar, eggs, and vanilla together until combined.
Stir in about half of the flour mixture. Then stir in the milk and the rest of the flour mixture and mix until fully combined. The batter will be thin.
Pour about ¼ cup of batter into each cupcake liner (they should be about halfway full).
Bake at 350°F for 18-22 minutes or until a cake tester comes out clean.
Remove from pan and let cool on wire rack. Let the cupcakes cool completely before decorating.
Use a cupcake corer to remove the center of each cupcake, then fill with a tablespoon of edible cookie dough.
Cookie Dough Buttercream
In a large bowl using a handheld or stand mixer fitted with a paddle attachment, cream the butter, brown sugar, and vanilla together until smooth.
Add in the powdered sugar and heat treated flour, and mix until fully combined. Beat the frosting on high for 4-5 minutes until light and fluffy.
Add in milk 1 teaspoon at a time until desired consistency is reached.
These cupcakes will last in the fridge for up to 3 days. However, they taste best when served at room temperature! I like to let them sit at room temperature for about an hour before serving.
Notes
Brown the butter: In a small saucepan, melt butter over medium heat. Once melted, stir frequently and watch closely for the formation of brown specs on the bottom of the pan and a nutty aroma. Once you see brown specs, stir for just a few more seconds (it browns super quickly). Remove from the heat and immediately pour into a different bowl to prevent it from burning. Let cool completely to room temperature. The brown butter should be solid but still pliable. Note: For every ½ cup (1 stick or 113g), you'll lose about 1 tablespoon (14g) to evaporation while cooking. So you can either add in an extra 1 tablespoon before browning, or if you're browning big batches, just make up the weight difference in your recipe with regular unsalted butter if needed.To make the heat treated flour: preheat the oven to 350°F. Spread 1 cup of all purpose flour on a parchment lined baking sheet. Bake for 5-7 minutes or until the flour registers 160°F on an instant read thermometer. Let cool completely before using.