In a mixing bowl, combine crushed cookies and melted butter. Lightly grease a 9-inch springform pan with butter and press the crumb mixture in it. (Instead of the vanilla cookies, a tasty variation that works well is 3/4 cup of graham cracker crumbs and 1/2 cup ginger snap cookie crumbs.).
In a mixing bowl, mix together cream cheese, brown sugar, corn starch, and cinnamon with an electric mixer. Add eggs and yolk and mix well. Purée banana in a food processor. Mix puréed banana, banana liqueur, rum, and vanilla extract into cheese mixture with mixer. Pour the cream cheese mixture over the crust.
Bake at 350 degrees F for 15 minutes.
Without opening the oven, lower the temperature to 225 degrees F and continue baking for 70 minutes, or until the center no longer jiggles when you gently shake the springform pan.
Remove the cake from the oven, close oven door, and gently run a knife all around the inside edge of the springform pan to prevent the cake from sticking to it and causing cracks. If cake is still a little wet in center, place it back in the oven for another 30 minutes with oven turned off. It will continue to cook gently as oven cools.
Remove cheesecake from oven and cool completely to room temperature.
Cover with plastic wrap and refrigerate overnight. Cooling to room temp prevents moisture from forming inside plastic wrap. Even cheesecakes that seem a little jiggly in the center will firm up nicely after refrigeration.