Preheat the oven to 356º F (180° C). Line a square baking pan or dish of about 9 inches (23 cm) with parchment paper.
In a medium bowl, sift together the flour, cocoa powder, salt and mix well with a spoon.
Separately, in a large bowl add the banana and mash it with a fork until you get a thick paste.
Add the sugar, butter and vanilla extract. Mix using a hand whisk or a fork for 1 minute until everything is well integrated.
Add the flour mixture to the bowl with the banana and beat for 1 more minute until smooth and lump-free. Then, stir in the chocolate chips until they are evenly mixed throughout the batter.
Pour the batter into the baking pan and spread it until it is evened out. Add spoonfuls of the peanut butter on top of the brownie mixture and then spread it all over the surface by drawing swirls using a wooden stick or a dull knife.
Bake at 356º F (180° C) for 25-30 minutes until a wooden toothpick comes out almost clean with just a few crumbs.
Remove the brownies from the oven and let them cool for at least 20 minutes before cutting them. Then, unmold them and cut them into 12 equal parts.