Heat oven to 350 degrees. Grease 3 6-inch cake pans liberally with butter or oil. In a small saucepot, melt butter over medium heat until solids begin to separate and butter turns a rich golden brown. Remove from heat and let cool 5-10 minutes.
In the bowl of a stand mixer or a medium bowl with a hand mixer, cream brown butter, yogurt, and sugar until fluffy, about 2-3 minutes. Whip in one egg at a time, then add vanilla and warm milk and mix thoroughly. In a separate bowl, combine flour, salt, and baking powder. Add in two batches to the wet ingredients and mix until mostly incorporated. Some streaks of flour are okay. Toss chocolate chips with 2 tbsp flour - this helps prevent the chips from sinking to the bottom of the cake layers - and fold into batter with a spatula.
Split batter evenly between the three cake pans. Bake 30-35 minutes until tops are set and a knife or toothpick comes out clean. Remove from oven and cool 5 minutes in the pans, then run a knife around the edges of the cakes to loosen and turn onto a wire rack. Cool completely.
While cakes bake, prepare the cookie dough. Cream peanut butter, butter, and sugars until soft, then blend in vanilla extract. Fold in flour. If the dough is crumbly, add milk or water 1 tbsp at a time until dough adheres. Cover with plastic wrap and set aside.
When cakes are ready, make the buttercream. With the stand mixer or a hand mixer, whip butter, peanut butter, and powdered sugar until soft and creamy.