1. Preheat the oven to 350°F. Grease a 9-inch by 13-inch pan and set aside.
2. In a saucepan over medium heat, melt the butter and brown sugar, stirring constantly until smooth. Remove from the heat, and stir in the peanut butter. Allow to cool while combining the flour, baking powder, baking soda and salt in a large mixing bowl.
3. Add the eggs, one at a time, mixing well after each addition. Add the vanilla extract and stir to combine. Pour the mixture into the prepared dry ingredients, and stir to create a thick, smooth batter. Fold in half the chocolate chips, reserving the rest for sprinkling on top.
4. Transfer the batter to the prepared pan, spreading the batter evenly to the edges of the pan. Sprinkle with the milk chocolate chips and lightly pressing them into the batter.
5. Bake for 25 minutes until the top is golden and the edges are slightly browned. Allow to cool in the pan. Cut into 1-2 inch squares and serve.