Preheat the oven to 350 degrees F (180 C). Generously butter an 8-inch** round springform pan. Line it with parchment paper, place it on a sheet pan, and set it aside.
Place butter in a small saucepan and heat until fully melted. Let it cool off the heat while you are working on the rest of the ingredients.
Core and peel apples, then cut them into 1-inch chunks. Set them aside.
Whisk together flour, baking powder, and salt in a small bowl. Set it aside.
In a large bowl, whisk together eggs and sugar until thoroughly mixed. Whisk in the rum and vanilla.
Whisk half of the dry ingredients into the wet ingredients, followed by half of the now-cooled melted butter.
Stir in the rest of the flour mixture and butter. Whisk until fully combined.
Fold in the apple chunks and make sure that they are fully coated with the batter.
Transfer the batter to the prepared cake pan and smooth the top with a spatula. Place the cake pan on a sheet pan (we do this to catch any bubbling juices.)
Bake for 50 minutes or until a toothpick inserted into the center comes out clean.
Let it cool for 5 minutes, then run a knife through the edges to loosen the cake from the pan.
Transfer to a cake platter (or a wire rack) and let it cool for at least an hour before serving. If preferred, you can sprinkle it with a small amount of powdered sugar before serving.