Make the dough: In a small saucepan over low heat, warm 1 cup of the milk and the butter until the butter is melted and the mixture is warm to the touch. Remove from the heat and transfer the mixture to the bowl of a stand mixer fitted with a dough hook. Sprinkle the yeast on top and let it sit until foaming, 8 to 10 minutes.
With the mixer on medium speed, add the sugar, salt, vanilla and 3 of the eggs, and mix until smooth. With the motor running, slowly add the flour and mix until just incorporated, 5 minutes. Increase the speed to medium high and beat until a dough ball forms, 4 to 6 minutes. Cover the bowl with plastic wrap and let sit in a warm place until the dough has doubled in size, about 1 hour.
While the dough is rising, make the filling: In a large skillet, heat the brown sugar and butter on medium-high heat until the sugar dissolves, 2 minutes. Add the cinnamon, rum and salt, and cook, stirring constantly, until the cinnamon is fragrant and the mixture has thickened, 2 minutes. Add the bananas to the skillet and gently toss to coat in the sauce, cooking for 2 minutes more. Transfer to a medium bowl and mix in the cream cheese and flour until smooth, breaking up the bananas as you mix. Let cool completely.
Preheat the oven to 350º and line a baking sheet with parchment paper. Divide the dough in half. On a lightly floured surface, roll out 1 piece into a 10-by-18-inch rectangle that's ¼ inch thick. Spread half of the banana mixture over the entire surface. Starting at one long end, roll the rectangle into a log. Repeat with the remaining dough and filling. Braid the two together and stretch the braid to 18 inches long. Connect the ends to form a 9-inch circle and transfer to the prepared baking sheet.
In a small bowl, whisk together the remaining tablespoon of milk with the remaining egg. Brush the top of the dough completely with the egg wash. Bake until golden brown, 40 minutes. Let cool completely, then insert a chocolate liquor bottle into the cake by making a small hole in the bottom.
Make the glaze: In a small bowl, whisk the milk, rum, confectioners' sugar and vanilla until smooth. Pour the glaze over the cake and decorate with sanding sugars. Let the glaze set slightly for 10 minutes, then slice and serve.