Position rack in center of oven and preheat oven to 350 degrees. Wrap outside of 9-inch diameter springform pan with 3-inch high sides with heavy duty foil. Combine flour, pecans, sugar, butter, brown sugar and vanilla in large bowl. Mix well, press mixture onto bottom of prepared pan.
Using an electric mixer, beat cream cheese in large bowl until smooth, gradually beat in sugar, beat in cornstarch. Add eggs 1 at a time, beating until blended after each addition. Add pureed banana, sour cream, lemon juice, vanilla, cinnamon and salt. Beat just until blended.
Transfer filling to crust-lined pan. Place pan in large roasting pan, add enough hot water to roasting pan to come up 1 inch up sides of springform pan. Bake until center of cake is just set, about 1 hour and 15 minutes. Remove cake from oven, maintain oven temperature.
Meanwhile, prepare topping:
Mix together sour cream, sugar and vanilla in a small bowl until well blended. Spread topping over cheesecake and place back in oven to bake until topping is set, about 10 minutes. Turn oven off, 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 and transfer cake to platter, warm caramel sauce in a small saucepan over low heat, stirring often, stir in rum. Drizzle sauce decoratively over cake and arrange bananas on top of cake.
Pour remaining topping in a small pitcher and drizzle over sliced cake before serving. You can also add a dollop of whipped cream J
Always use caution when working around the kitchen, stove and cutlery.