Preheat the oven to 350°F. Grease a 10×5-inch loaf pan and line with parchment paper. Set aside.
Following the instructions for how to brown butter, brown the butter in a saucepan over medium heat. When browned, remove from the heat and pour into a bowl to cool while you work on the batter.
In a large bowl, whisk together the granulated sugar, flour, baking powder, baking soda, salt, and nutmeg.
In a separate bowl, whisk together the mashed bananas, eggs, and vanilla extract. Pour into the dry ingredients and mix until just combined.
Add the cooled brown butter and stir until just incorporated. Pour the batter into the prepared loaf pan.
To make the streusel, combine the brown sugar, flour, cinnamon, and salt in a small bowl. Use a fork or your fingers to work the butter into the mixture until clumps form. Sprinkle evenly over the bread batter.
Bake for 1 hour 10 minutes, or until a toothpick inserted into the center comes out clean. Let cool for 15 minutes in the pan, then transfer to a wire rack to cool completely.
Notes
Store, well wrapped, at room temperature for up to 3 days or in the refrigerator for up to 5 days.