Preheat the oven to 350°F (180°C). Butter and line an 8x8-inch baking pan with parchment paper.
In a medium bowl, whisk together the flour, baking powder, and salt.
In a separate large bowl, whisk the sugar, melted butter, eggs, and vanilla extract until smooth.
Fold the dry ingredients into the wet ingredients until just combined.
Divide the batter evenly between two bowls.
In the first bowl, sift in the cocoa powder and stir in the chopped bittersweet chocolate.
In the second bowl, fold in the mashed bananas until fully combined.
Spoon the chocolate brownie batter into the prepared pan, then pour the banana batter over the top.
Use a butter knife to gently swirl the two batters together to create a marbled effect.
Bake the Banana Bread Brownies for 40–45 minutes, or until a toothpick inserted in the center comes out with a few moist crumbs.
Let cool completely before slicing into squares.