Line a square 8x8 baking pan with parchment paper and preheat the oven to 350°F/180°C.
Cook the milk, sugar, flour, and egg yolk in a small pot over medium-low heat. Stir constantly until it's thick like pudding, about 5-8 minutes. Pour the pudding into a bowl and stir in the butter and vanilla. Set aside.
In a large bowl, whisk together the melted butter, brown sugar, and white sugar. Then whisk in the egg yolk, mashed banana, and vanilla extract.
Fold in the flour, baking powder, and salt until just combined. Then evenly spread half of the batter into the prepared baking pan.
Press the vanilla wafer cookies on top of the batter. Then use a spoon to alternate dollops of the reserved banana batter and vanilla pudding on top. Use a knife or toothpick to swirl the batter and pudding together, then press more vanilla wafers into the top.
Bake for 30-35 minutes or until a toothpick inserted into the center comes out clean. Let the pan of brownies cool completely on a wire cooling rack then cut into 16 small brownies or 9 large ones.