In medium mixing bowl beat brown sugar, egg yolks, and cinnamon with an electric mixer about 4 minutes; set aside.
In medium saucepan heat and stir the milk, cream, and 1/2 teaspoon salt until just simmering. Slowly stir about 1 cup of the hot milk mixture into the egg yolk mixture. Return all egg yolk mixture to saucepan. Heat and stir constantly (do not boil) until mixture thickens. Remove pan from heat and place in bowl of ice water; stir 2 to 3 minutes to cool.
Mash the 2 bananas with the lemon juice. Stir into thickened mixture along with rum and vanilla (mixture may appear slightly curdled). Transfer to large bowl. Cover; refrigerate overnight.
Freeze in 1-1/2- or 2-quart ice cream freezer according to manufacturer's directions. Serve with sliced bananas and caramel topping. Makes about 8 (1/2-cup) servings.