For crust, preheat oven to 350 degrees F. and grease the bottom of a 9-inch springform pan and wrap the pan in foil.
In a food processor, pulse oatmeal cookies with pecans until an even crumble. Add melted butter and pulse to combine. Press into the bottom of prepared springform pan (not up the sides) and bake for 12 minutes. Allow to cool while preparing bananas.
For bananas, heat a sauté pan over medium-high heat and add butter and brown sugar, stirring until melted and bubbling. Stir in vanilla and rum and add bananas. Stir just to coat and warm and pour into cooled crust.
For cheesecake, beat cream cheese until fluffy, scraping down the sides of the bowl often. Add brown sugar in 2 additions, beating well after each addition. Beat in butter. Add eggs, one at a time, and beat well after each addition. Stir in vanilla and rum extract and pour cheesecake batter over caramelized banana layer in springform pan.
Bake cheesecake in a water bath: place springform pan into a baking dish and pour hot tap water around it, to come halfway up the pan. Bake for 30 minutes, then reduce oven temperature to 325 degrees F. and bake for another 30 to 40 minutes, until filling puffs just slightly around edges but still moves a little in center when shaken.
Remove pan from water and allow to cool for 2 hours before refrigerating overnight.
For caramel topping, combine sugar, water and corn syrup or lemon juice in a pan. Bring to a boil without stirring and cook, uncovered, until it turns a rich amber colour. While cooking, occasionally brush down the sides of the pot with a brush dipped in cool water.
Once sugar has reached desired colour, remove from heat and carefully stir in whipping cream (watch out for the steam and bubbling). Return to medium heat and simmer until reduced by a third, stirring occasionally, about 5 minutes. Chill until thickened but still fluid, about 15 minutes. Pour caramel over cheesecake and chill completely. (This can be done up to 8 hours before serving).
To serve, run a knife carefully along the inside of the pan to loosen, remove pan and slice with a hot, dry knife.