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 match. Shake the pan back and forth until the flame dies, basting the bananas continuously. Remove from the heat and cool completely. In a food processor, fitted with a metal blade, add the banana mixture and puree until smooth. In another saucepan, add the cream. Bring the liquid to a gentle boil. Whisk the egg yolks together.
Whisk 1 cup of the hot cream into the egg mixture. Mix thoroughly. In a steady stream, slowly add the egg mixture to the hot cream mixture. Continue to cook for 4 minutes, stirring occasionally. Remove from the heat and cool completely. Whisk the banana mixture into the cream mixture. Mix thoroughly and cool completely. Add the banana/cream mixture to the electric ice cream machine. Process according to manufactures directions. Place a tablespoon of the caramel over each tart shell. Spread a heaping 1/2 cup of the ice cream over the caramel sauce, filling each tart shell completely. Place the tarts in the freezer until frozen, about 1 hour. In the bowl of an electric mixer, fitted with a wire whip, whip the egg whites until soft peaks form. Add the sugar and continue to beat to stiff peaks. Remove the tarts from the freezer and spread the meringue over ice cream. Place the tarts on a baking sheet and place in the oven. Bake until the meringue is golden brown, about 4 to 5 minutes. Remove from the oven and serve.