Prepare your pan and oven: Preheat oven to 325°F (165°C). Wrap the outside of a 9-inch springform pan with aluminum foil to prevent water leakage if using a water bath. Grease the inside of the pan lightly.
Make the crust: In a medium bowl, combine graham cracker crumbs, sugar, and salt. Pour in melted butter and mix until the mixture resembles wet sand. Press firmly into the bottom and about 1 inch up the sides of your prepared pan. Use the bottom of a measuring cup for even packing.
Pre-bake the crust: Bake for 10 minutes until lightly golden. Remove and let cool while preparing the filling. Keep the oven on.
Beat the cream cheese: In a large bowl, beat the softened cream cheese and sugar with an electric mixer on medium speed until completely smooth and fluffy, about 4-5 minutes. Scrape down the sides of the bowl frequently to ensure no lumps remain.
Add eggs one at a time: Beat in eggs one at a time on low speed, mixing just until each is incorporated. Overbeating after adding eggs can cause cracks in your finished cheesecake.
Fold in remaining ingredients: Add vanilla extract, sour cream, and heavy cream. Mix on low speed just until combined and smooth. The batter should be silky and somewhat thick.
Pour and bake: Pour the filling into the cooled crust. If using a water bath (recommended for crack prevention), place the foil-wrapped springform pan in a larger roasting pan and add hot water to reach halfway up the sides of the springform pan. Bake for 55-65 minutes until the edges are set but the center still has a slight jiggle.
Cool properly: Turn off the oven, crack the door open, and let the cheesecake cool in the oven for 1 hour. This gradual cooling helps prevent cracks. Remove from oven and cool completely at room temperature.
Prepare the pineapple topping: While the cheesecake cools, combine pineapple, sugar, cornstarch, water, and lemon juice in a medium saucepan. If using fresh pineapple, first pulse chunks briefly in a food processor to break them down slightly. Bring mixture to a simmer over medium heat, stirring frequently until thickened, about 5-7 minutes. Remove from heat and stir in rum or extract if using. Let cool completely.
Assemble and chill: Spread the cooled pineapple topping over the cheesecake. Refrigerate for at least 4 hours, preferably overnight, before serving.
Serve: Run a thin knife around the edge of the pan before releasing the springform ring. Slice with a clean, warm knife, wiping the blade between cuts for neat slices.