Preheat the oven to 350°F (175°C). Generously butter two 8 by 4-inch (20 by 10-cm) or 9 by 5-inch (23 by 13-cm) loaf pans, then line them with buttered parchment paper. In a large bowl, combine the bananas, brown sugar, butter, milk, eggs, and vanilla, and mix well with a wooden spoon.
In a separate bowl, whisk together the flour, baking powder, baking soda, and salt. Add the dry ingredients to the banana mixture and stir just until smooth.
Scoop the Nutella into a small bowl and heat in the microwave until softened enough to drizzle, 20 to 30 seconds.
Divide half of the banana batter between the prepared pans. Drizzle half of the Nutella over the two pans and run a small, thin knife through the batter in a zigzag or swirl pattern to distribute.
Add the remaining batter to the pans, top with the remaining Nutella, and swirl into the batter with the knife again.
Bake until a tester comes out clean, 40 to 50 minutes. Let the cakes cool in the pans for 10 minutes, then remove from the pans and set on a wire rack to cool completely before slicing.