In a bowl, whisk together warm milk, yeast, and a pinch of sugar. Allow the mixture to sit until it becomes foamy, which should take about 5 minutes.
Incorporate mashed bananas, melted butter, egg, granulated sugar, and salt into the yeast mixture. Stir the ingredients together until well combined.
Gradually mix in the flour until a smooth dough is formed. Knead the dough for 5 minutes, then cover it and let it rise until it has doubled in size, approximately 1 hour.
Roll the dough out into a rectangle shape. Spread the softened butter over the top, then sprinkle the brown sugar and cinnamon evenly. Roll the dough up tightly and slice it into 9 individual rolls.
Arrange the rolls in a greased baking dish. Cover the dish and allow the rolls to rise for an additional 20 minutes. Bake in a preheated oven at 350°F (175°C) for 25 minutes, or until they turn golden brown.
In a separate bowl, whisk together the powdered sugar and milk to create the icing. Drizzle the icing over the warm rolls just before serving.