Heat butter in a skillet over medium heat until browned. Transfer to a bowl and set aside.
In a large bowl, combine warm milk and yeast. Let it rest until yeast blooms, about 5 minutes.
Add mashed banana, maple syrup, egg, brown butter, and vanilla. Whisk until combined.
Using a stand mixer with a dough hook, gradually stir in flour and salt until a smooth dough forms.
Transfer dough to an oiled bowl, cover with plastic wrap, and let rise in a warm place for 1-1.5 hours.
Brown the butter for the icing in a skillet over medium heat. Let it cool.
In a stand mixer, blend cream cheese, vanilla, and browned butter until smooth.
Gradually add powdered sugar, beating until fluffy. Set aside.
Roll out the risen dough into a rectangle on a floured surface.
Spread softened butter over dough, sprinkle with filling, and arrange banana slices.
Roll dough into a log, then slice into rolls and place in a baking dish.
Let rolls rise for 30 minutes in a warm place.
Preheat oven to 350°F and bake rolls for 22 minutes.
Allow to cool for 10 minutes before frosting with the brown butter icing.