Step 1: Activate the yeast. In a small bowl, combine ½ cup warm milk and 2¼ teaspoons active dry yeast. Let stand for 5-10 minutes until foamy.
Step 2: Combine wet ingredients. In a large mixing bowl, mash 1 cup ripe bananas until smooth. Stir in ¼ cup granulated sugar, ¼ cup melted butter, 1 teaspoon salt, and 1 beaten egg until well combined.
Step 3: Incorporate yeast mixture. Add the foamy yeast-milk mixture to the banana mixture and stir until just combined.
Step 4: Knead the dough. Gradually add 3 cups all-purpose flour, stirring until a shaggy dough forms. Turn out onto a lightly floured surface and knead for 6-8 minutes, adding more flour as needed, until the dough is smooth and elastic.
Step 5: First rise. Place the dough in a lightly greased bowl, cover with plastic wrap, and let rise in a warm, draft-free place for 1-1.5 hours, or until doubled in size.
Step 6: Prepare the filling. In a small bowl, combine ½ cup brown sugar and 1 tablespoon ground cinnamon.
Step 7: Roll and fill the dough. On a lightly floured surface, roll the risen dough into a 12x16 inch rectangle. Spread 3 tablespoons softened butter evenly over the dough, then sprinkle the cinnamon-sugar mixture on top, pressing lightly.
Step 8: Shape the rolls. Starting from one long edge, tightly roll the dough into a log. Pinch the seam closed. Using a sharp knife or unflavored dental floss, cut the log into 12 equal rolls, approximately 1½ inches each.
Step 9: Second rise. Arrange the rolls, cut-side up, in a greased 9x13 inch baking dish, leaving a small gap between each. Cover loosely and let rise in a warm spot for 30-45 minutes, until puffy and nearly doubled. Preheat oven to 350°F (175°C) during this time.
Step 10: Bake the rolls. Remove the cover and bake for 20-25 minutes, or until the tops are golden brown and the centers spring back slightly. If edges brown too quickly, tent with foil.
Step 11: Prepare the icing. While rolls cool slightly, beat together 4 ounces softened cream cheese, 2 tablespoons softened butter, 1 cup powdered sugar, and ½ teaspoon vanilla extract in a small bowl. Add 1-2 tablespoons of milk, a little at a time, until a smooth, pourable consistency is reached.
Step 12: Cool and ice. Let the rolls cool in the pan for 5-10 minutes. Drizzle or spread the cream cheese icing over the warm rolls just before serving.