In a medium-size skillet over medium heat, melt 3 tablespoons of the butter. Add the sugar and stir until it dissolves. Add the bananas and cook, stirring, for 3 minutes. Remove the pan from the stove and carefully add the liqueur and rum. Return the pan to the stove. Using long matches and keeping long hair and hanging sleeves well clear of the pan, carefully ignite the alcohol and stir with a long-handled wooden spoon until the flames die down, about 30 seconds. Remove from the heat and let cool completely. Purée the mixture in a food processor or blender.
In a medium-size nonreactive saucepan over medium heat, heat the milk and cream to the scalding point (when bubbles form around the edges of the pan). Do not let it boil. Remove from the heat.
Beat the egg yolks in a large mixing bowl. Add the cream mixture about 1/4 cup at a time to the beaten eggs, whisking well after each addition. Pour the mixture back into the saucepan, and cook, stirring, over medium heat until the custard becomes thick enough to lightly coat the back of a spoon, 2 to 3 minutes. Do not boil.
Remove from the heat and pour the mixture into a large glass mixing bowl. Add the puréed bananas and stir to mix. Cover the top of the mixture with plastic wrap pressed against the surface (this will keep a skin from forming) and let cool. Place the mixture in the refrigerator and chill completely.
Pour the banana mixture into an ice cream machine and follow the manufacturer’s instructions for the churning time. If you leave the ice cream in the freezer for a while before completing the pie, be sure to let it soften up a bit first.
Make caramel sauce: In a small heavy saucepan, combine the sugar and water and bring to a boil, stirring often. Cook, stirring occasionally, until the mixture is a deep caramel color and has the consistency of thick syrup, 10 to 15 minutes. Remove from the heat. Stir in the cream, return the saucepan to high heat, and boil the sauce until it regains the consistency of a thick syrup, about 2 minutes. Let cool. The sauce can be refrigerated until ready to use. Allow it to reach room temperature before using it. Makes about 3/4 cup.
Preheat the oven to 375 degrees F.
Put the graham cracker crumbs in a medium-size mixing bowl. Melt the remaining 5 tablespoons butter and add to the crumbs. Mix well. Press evenly into the sides and bottom of a 9 1/2-inch deep-dish pie pan and bake until browned, about 10 minutes. Let cool completely.
Carefully spread the caramel sauce evenly over the inside of the graham cracker shell. Fill the shell with the ice cream, spreading it evenly with a rubber spatula. Cover with plastic wrap and freeze until the ice cream firms up, about 4 hours. Cut the pie into wedges to serve.