Position rack in center of oven and preheat oven to 350° F.
Wrap outside of 9-inch diameter springform pan with 3-inch-high sides with heavy-duty foil.
Combine flour, pecans, butter, sugar, brown sugar and vanilla extract in large bowl. Mix well. Press mixture onto bottom of prepared pan.
Using electric mixer, beat cream cheese in large bowl until smooth. Gradually beat in sugar. Beat in cornstarch. Add eggs, one at a time, beating until just blended after each addition. Add puréed banana, sour cream, lemon juice, vanilla extract, cinnamon and salt. Beat just until combined.
Transfer filling to crust-lined pan. Place pan in large roasting pan. Add enough hot water to roasting pan to come one inch up sides of springform pan.
Bake until center of cake is just set, about 1 hour 15 minutes. Remove cake from oven. Maintain oven temperature.
Meanwhile, prepare topping. Mix together sour cream, sugar and vanilla extract in small bowl until well blended. Spread topping over cheesecake. Bake until topping is set, about 10 minutes. Turn off oven. Let cake stand in oven until cooled to room temperature, about 2 hours.
Refrigerate cake until well chilled. Cover and refrigerate overnight. Cut around cake to loosen; remove pan sides. Transfer cake to platter. Warm caramel sauce in small saucepan over low heat, stirring often. Mix in rum. Drizzle some sauce decoratively over cake. Arrange bananas atop cake. Cut cake into wedges and serve with sauce.Servings: 10 to 12