Preheat oven to 350. Butter and flour a 9-inch springform pan.
Peel apple and chop into roughly ¼-inch dice.
Whisk together flour, salt, and baking powder in a small bowl.
In a larger bowl, combine the softened butter, white sugar, and brown sugar. Using a hand mixer or stand mixer, cream together until light and fluffy.
Add the eggs one at a time, mixing thoroughly until well combined. Then mix in the vanilla extract and rum.
Stir in the flour mixture until just combined, then fold in the chopped apples.
The batter will be very thick, almost like cookie dough.
Pour the batter into the springform pan and spread evenly.
Bake 25-30 min, until a cake tester or toothpick comes out almost clean, and the top is nicely browned.
Let the cake cool in the pan for a short while (10–15 mins), then remove the outside ring and finish cooling on a rack or serve slightly warm.
Dust with powdered sugar or a dollop of whipped cream to garnish, if desired.
Notes
Storage Tips: Store cake in springform pan covered with aluminum foil, or cut into slices and store in an airtight container. Cake will last for 3 days. For longer storage, keep the cake in the refrigerator, or wrap individual pieces in plastic wrap and foil, then freeze for up to 2 months. Bring back to room temperature to serve.Expert Tips: Don’t over-mix once you add the flour; you want the cake to have a tender crumb, not tough.¼-inch pieces for your apples give you enough structure so the pieces stay distinct, but still tender. You can thinly slice your apples if you prefer, but I like having little chunks of apple to bite into.Baking time can vary depending on your oven and pan size — start checking at 25 mins if your oven tends to run hot.Make sure to let the cake rest a little before slicing so that it can settle and firm up. That makes for prettier slices.When you’re ready to serve, powdered sugar or whipped cream make a pretty topping. You can also serve with a scoop of vanilla ice cream.