Prepare the Pecan Shortbread Crust and set aside to cool.
In a large saute pan, melt the butter. Stir in the brown sugar and cinnamon. Add the bananas and baste them with the syrup for 2 minutes. Pull the saute pan off the burner. Carefully add the banana liqueur and rum. Place the pan back on the stove and shake the pan a couple of times. The pan should flame, if not, ignite the pan with a long match. Shake the pan back and forth until the flame dies, basting the bananas continuously with a long handled spoon. Remove from the heat and cool completely.
In a food processor fitted with a metal blade, add the banana mixture and puree until smooth. Set aside.
In another saucepan, add the cream and bring to a gentle boil. In a mixing bowl, whisk the egg yolks and granulated sugar together until pale yellow. Whisk 1 cup of the hot cream into the egg mixture and mix thoroughly. In a steady stream, slowly add the egg-cream mixture to the hot cream remaining in the saucepan. Continue to cook for 4 minutes, stirring occasionally, until the sauce thickens enough to coat the back of a spoon. Do not allow the mixture to boil. Remove from the heat and cool completely. Whisk the banana puree into the cream mixture and mix thoroughly. Transfer to the refrigerator and chill thoroughly. Process in an electric ice cream machine according to manufacturers' instructions.
When the ice cream has frozen to the desired consistency, transfer to the springform pan with the cooled shortbread crust and, using a spatula, spread to form a smooth surface. Wrap with plastic wrap and place in the freezer for at least 4 hours and up to 4 days.
When ready to serve, transfer the ice cream pie to the refrigerator for 20 minutes or so to thaw a little so that it is easier to slice. Serve slices drizzled with the caramel sauce.
Lightly butter the bottom of an 8 or 9-inch springform pan. Set aside.
In the bowl of an electric mixer, cream the butter and sugar on high speed until light and fluffy, 3 to 4 minutes. Sift the flour and salt into a medium-mixing bowl. With the mixer on low speed, add the flour mixture to the creamed butter mixture and continue mixing only until the flour is just incorporated. Add the nuts and mix until just combined.
Using your fingers, press the dough so that it forms a smooth layer covering the entire surface of the springform pan bottom and slightly up the sides. Using the tines of a fork, prick the pastry all over at regular intervals, pressing all the way through to the bottom. Press the tines of the fork all around the outer edges of the pan. Transfer the pan to the refrigerator to chill for at least 30 minutes before baking.
Preheat the oven to 300 degrees F.
Bake the shortbread until it is pale golden, 35 to 40 minutes. Cool the shortbread on a wire rack until completely cool.
Combine the sugar, water, and lemon juice in a medium heavy saucepan and cook over medium-high heat, stirring, until the sugar dissolves. Let boil without stirring until the mixture becomes a deep amber color, 2 to 3 minutes, watching closely so it doesn't burn. Add the cream (be careful; it will bubble up), whisk to combine, and remove from the heat.
Add 2 tablespoons of the milk, then add up to 2 more tablespoons, until the desired consistency is reached. Let cool until just warm before serving. (The sauce will thicken as it cools.)