Make the dough: In a large mixing bowl, combine the warm water, yeast, and sugar. Let it sit for 5-10 minutes, until the yeast is activated and foamy.
Add dry ingredients: Add the flour, salt, and melted butter to the bowl. Mix until a shaggy dough forms.
Add egg and bananas: Add the egg, mashed bananas, and vanilla extract to the bowl. Mix until the dough comes together in a sticky ball.
Knead the dough: Knead the dough on a floured surface for 10-12 minutes, until it becomes smooth and elastic.
Let it rise: Place the dough in a greased bowl, cover it with plastic wrap, and let it rise in a warm place for 1 hour, until it has doubled in size.
Prepare the filling: In a small bowl, mix together the granulated sugar, cinnamon, and softened butter until well combined. Stir in the mashed bananas.
Roll out the dough: Roll out the dough on a floured surface to a 1/4-inch thickness.
Spread the filling: Spread the filling evenly over the dough, leaving a 1-inch border around the edges.
Roll the dough: Roll the dough into a tight log, starting from the long side. Cut the log into 12-15 equal pieces.
Let it rise again: Place the rolls on a baking sheet lined with parchment paper, leaving about 1 inch of space between each roll. Cover with plastic wrap and let it rise for another 30-40 minutes, until it has puffed up slightly.
Bake: Preheat the oven to 375°F (190°C). Bake the rolls for 18-20 minutes, until golden brown.
Glaze: In a small bowl, whisk together the powdered sugar, milk, and vanilla extract. Drizzle the glaze over the warm rolls.