In a medium sauté pan, melt butter over medium heat. Add brown sugar and cinnamon; cook 3 to 4 minutes, stirring constantly. Carefully add rum, and cook 2 minutes, stirring frequently.
Add bananas, stirring gently to coat. Remove from heat; cool mixture slightly.
Spoon about 3 tablespoons mixture in center of each tortilla. Fold in sides and roll up. Secure with toothpicks.
Fill a Dutch oven with two inches of oil; heat to 350°. Fry two filled tortillas at a time 1 to 2 minutes, or until golden brown. Drain on paper towels. Sprinkle with cinnamon sugar.
To serve, top warm chimichangas with vanilla ice cream, caramel-flavored topping, and chopped pecans, if desired.