Preheat the oven to 325°F (160°C). Prepare a 9-inch springform pan by lightly greasing it.
Make the crust: In a bowl, mix graham cracker crumbs, melted butter, and sugar until evenly combined. Press firmly into the bottom of the springform pan. Bake for 10 minutes, then set aside to cool.
Prepare the cheesecake filling: In a large bowl, beat cream cheese and sugar until smooth and creamy. Add eggs one at a time, mixing well after each addition. Fold in heavy cream and vanilla extract until fully combined.
Pour the filling over the crust and smooth the top with a spatula.
Bake the cheesecake: Place the springform pan in a water bath (wrap the pan bottom with foil to prevent leaks), and bake for about 60 minutes or until the edges are set but the center slightly jiggles. Turn off the oven and leave the cheesecake inside with the door slightly open for 30 minutes to cool gradually.
Cool completely at room temperature, then refrigerate for at least 4 hours or overnight.
Caramelize the bananas: In a skillet, melt butter over medium heat. Add brown sugar and stir until dissolved and bubbling. Add banana slices and cook gently for 2–3 minutes until golden and syrupy, stirring carefully to avoid breaking the bananas. Remove from heat and let cool slightly.
Prepare the rum glaze: In a small saucepan, combine rum and sugar. Bring to a simmer over medium heat and cook for 5–7 minutes until the mixture thickens slightly. Remove from heat and stir in butter until smooth.
Assemble the cheesecake: Arrange caramelized bananas over the chilled cheesecake. Drizzle the rum glaze generously on top.
Chill again for 30 minutes to let the glaze set. Serve cold and enjoy!