Preheat oven to 350°F (175°C) and grease a 9x13-inch pan.
Mash bananas in a bowl until smooth.
In a separate bowl, mix flour, baking soda, baking powder, and salt.
In another bowl, beat eggs with sugar and brown sugar.
Add melted butter, milk, vanilla, and mashed bananas; mix well.
Combine wet and dry ingredients; stir until just blended.
Fold in chopped nuts.
Pour batter into prepared pan and smooth the top.
Bake 35–40 minutes or until a toothpick comes out clean.
Cool on a rack before slicing; optional frosting if desired.