Preheat oven to 350o F. Line or grease an 8x8 inch baking pan.
Stir the flour, baking powder, and salt together in a bowl.
In a medium sauce pot over medium heat, melt butter. Continue to cook while it foams and stir constantly until it browns and you see some brown specks at the bottom and it starts to smell a bit toasted. Remove from heat. Once butter cools a bit, add sugar and vanilla extract. Next, beat in the eggs one at a time with a spoon. Gradually add the flour mixture, and stir the batter until it's evenly moistened. Divide the batter in half to two separate bowls.
Into one bowl of batter, blend the cocoa powder and chocolate chips. This mixture will be very thick. In the second bowl of batter, stir in the mashed banana.
Spread 1/2 of the chocolate batter into the bottom of the prepared baking pan, and follow with all of the banana batter. For the remaining chocolate batter, you can either spoon it on top or you can put it in a plastic bag, snip off the corner, and squeeze it onto the banana batter in lines. Use a butter knife or a spatula to swirl the two batters together to create a marbled appearance.
Bake in the preheated oven until the brownies begin to pull away from the sides of the pan, and a toothpick inserted into the center comes out clean, 40 to 45 minutes (mine took exactly 40 minutes and they were perfect). Cool in the pan, cut into squares, and serve.