Preheat the oven to 350°F (175°C). Generously butter an 8-inch round springform pan. Place on a parchment-lined rimmed baking sheet.
Whisk together the flour, baking powder, and salt. Set aside.
Beat the eggs with a whisk until they’re foamy. Add the sugar, and whisk for about a minute or until combined. Next, whisk in the rum and vanilla.
Add half the flour, whisking to combine. Then gently mix in half of the melted butter, followed by the remaining flour, and finally the remaining butter. The batter should be smooth and thick.
Fold the apples into the batter, making sure to coat them well.
Transfer the batter to the prepared pan, and spread evenly. Bake for 50 to 60 minutes, or until the top is golden brown and a thin knife inserted into the center comes out clean.
Place the pan on a cooling rack, and allow it to cool for 5 minutes. Run a knife around the edges of the cake, and then remove the sides of the pan.
Serve warm or at room temperature.
Notes
*For the best flavor, use a variety of apples. Gala, Pink Lady, Fuji, Braeburn, Honeycrisp, Granny Smith, and Jonagold are all great choices.