Make the bun dough
Place the milk in a microwave-safe mug and microwave on high until warm (about 110°F), 30–35 seconds. It should feel like very warm bath water, not scalding. Add the milk to the bowl of a stand mixer, scatter the yeast on top, then add the sugar, egg, and melted butter.
Attach the dough hook and mix on low speed until the ingredients begin to come together, about 15 seconds. Increase the speed to medium-low, gradually add the flour, and mix for 3 minutes. Add the salt and continue mixing until the dough is slightly tacky but pulls away from the sides and bottom of the bowl, about 5 minutes. Lightly grease a medium bowl, transfer the dough to the bowl, cover with plastic wrap, and let rise in a warm place until doubled in size, about 1 hour and 15 minutes.
Make the brown sugar sprinkle
While the dough is rising, combine the brown sugar, 2 teaspoons cinnamon, nutmeg, and ⅛ teaspoon salt in a small bowl.
Make the banana filling
Heat 5 tablespoons of the butter in a 9- or 10-inch skillet over medium-high heat, stirring often, until foamy and browned with nutty flecks, about 4 minutes. Scoop out 2 tablespoons of the brown butter and reserve in a small heatproof bowl for the frosting.
Add the bananas, sugar, bourbon, remaining teaspoon cinnamon, and remaining ⅛ teaspoon salt to the skillet. Cook, shaking and stirring occasionally, until the liquid thickens slightly, 3–4 minutes. Transfer the banana mixture to a plate and let cool to room temperature.
Fill and form the rolls
Line the bottom and sides of a 13x9-inch baking pan or large cast-iron skillet with parchment paper. Lightly flour a work surface and turn the risen dough out onto it.
Gently shape the dough into a rectangle, then roll it out into a 9x14-inch rectangle. Spread the remaining 3 tablespoons room-temperature butter evenly over the dough. Sprinkle the brown sugar mixture over the butter and press gently. Spoon the banana mixture evenly over the top, pressing lightly into the dough.
Starting from a short end, roll the dough into a tight log, gently pulling the sides outward as you roll to keep it even. Trim ½ inch from each end. Mark the log into 12 equal sections (about 1 inch each), then use a lightly floured serrated knife to gently saw through the dough, creating 12 rolls. Arrange the rolls in the prepared pan, cover with plastic wrap, and let rise until puffy, about 45 minutes.
Preheat the oven to 350°F. Remove the plastic wrap and bake until the rolls are fluffy and golden on top, about 25 minutes.
Make the frosting
While the rolls bake, combine the cream cheese, reserved 2 tablespoons brown butter, confectioners’ sugar, and salt in the bowl of a stand mixer fitted with the paddle attachment. Beat on low speed for 30 seconds, then increase to high and beat for another 30 seconds. Add the milk and bourbon and beat until light and fluffy, 2–3 minutes.
Remove the rolls from the oven, let cool for 5 minutes, then spread the frosting evenly over the warm rolls.
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