Preheat oven to 350°F (175°C). Grease an 8-inch square baking pan.
Whisk together the flour, baking powder, salt, and baking soda. Set aside.
In a separate bowl, combine the cooled butter, brown sugar, egg, and vanilla until thoroughly combined.
Add the flour mixture, mixing just until combined. Mix in the bananas.
Transfer the batter to the prepared pan and spread evenly.
Dollop the dulce de leche over the top of the batter. Use a thin knife to swirl the dulce de leche into the batter. (Alternatively, you can mix it into the batter before transferring to the pan.)
Bake 35 to 40 minutes, or until a pick inserted into the center comes out clean.
Cool in the pan before cutting into bars.
Notes
*Look for dulce de leche alongside other spreads, in the international foods section, or with the ice cream toppings.
Store in an airtight container at room temperature up to 3 days.