Preheat oven to 350 degrees F (175 degrees C). Lightly grease a 9x13-inch baking pan.
Beat 3/4 cup butter, white sugar, and brown sugar together in a bowl using an electric mixer until light and fluffy; beat in eggs and vanilla extract.
Mix 2 1/4 cup flour, baking powder, and salt together in a separate bowl; stir flour mixture into creamed butter mixture until batter is smooth. Divide batter into two bowls.
Whisk cocoa powder and 1 tablespoon melted butter together in a bowl; stir into 1 of the bowls of batter. This will be the 'black half'. Pour black half into the prepared baking pan.
Stir the remaining 1/4 cup flour and milk chocolate chips into the remaining bowl of batter creating the 'white half'. Spread white half evenly over black half. Sprinkle white chocolate chips over white half.
Bake in the preheated oven until edges begin to pull away from the sides of the pan, 20 to 30 minutes. Cool completely before cutting into bars.