Prepare the Pan: Preheat the oven to 350°F (180°C or 170°C fan-forced). Line a 9 × 13-inch (23 × 33 cm) baking pan with parchment paper or aluminum foil, leaving some overhang around the sides for easier removal, or lightly grease the pan.
Melt the Butter: Add the butter and sugar to a large bowl. Microwave for 45 seconds on medium power, then remove and stir. Continue repeating this process until the butter has melted. Alternatively, melt the butter and sugar using a double boiler. Cool the Mixture: Allow the melted butter and sugar mixture to cool slightly before adding the eggs, preventing the mixture from being excessively hot when they are incorporated.
Add the Eggs: Whisk the eggs and vanilla extract into the slightly cooled butter mixture until incorporated.
Incorporate the Banana: Stir the mashed bananas into the mixture until evenly distributed throughout the batter.
Combine the Dry Ingredients: If the cocoa powder is lumpy, sift it first. Gently whisk the flour, cocoa powder, and salt into the banana mixture until combined. The batter may still contain some lumps from the mashed banana.
Add the Chocolate Chips: Stir the chocolate chips through the batter until they are distributed throughout.
Fill the Pan: Spoon or pour the brownie batter into the prepared 9 × 13-inch pan, spreading it into the pan.
Bake the Brownies: Bake at 350°F (180°C or 170°C fan-forced) for about 35 minutes, or until a toothpick inserted into the brownies comes out clean or with a few damp crumbs.
Cool Completely: Let the brownies cool fully, which takes about 4 hours, before slicing them into portions.
Store the Brownies: Place the cooled brownies in an airtight container and refrigerate for up to 4 days.