Center a rack in the oven and preheat the oven to 350°F Generously butter an 8-inch springform pan and put it on a baking sheet lined with a silicone baking mat or parchment paper and put the springform on it.
Whisk the flour, baking powder, and salt together in small bowl.
Peel the apples, cut them in half and remove the cores. Cut the apples into 1- to 2-inch chunks.
In a medium bowl, beat the eggs with a whisk until they’re foamy. Pour in the sugar and whisk for a minute or so to blend. Whisk in the rum and vanilla. Whisk in half the flour and when it is incorporated, add half the melted butter, followed by the rest of the flour and the remaining butter, mixing gently after each addition so that you have a smooth, rather thick batter. Switch to a rubber spatula and fold in the apples, turning the fruit so that it's coated with batter. Scrape the mix into the pan and poke it around a little with the spatula so that it's evenish.
Slide the pan into the oven and bake for 50 to 60 minutes, or until the top of the cake is golden brown and a knife inserted deep into the center comes out clean; the cake may pull away from the sides of the pan. Transfer to a cooling rack and let rest for 5 minutes.
Carefully run a blunt knife around the edges of the cake and remove the sides of the springform pan. (Open the springform slowly, and before it’s fully opened, make sure there aren't any apples stuck to it.) Allow the cake to cool until it is just slightly warm or at room temperature. If you want to remove the cake from the bottom of the springform pan, wait until the cake is almost cooled, then run a long spatula between the cake and the pan, cover the top of the cake with a piece of parchment or wax paper, and invert it onto a rack. Carefully remove the bottom of the pan and turn the cake over onto a serving dish.
STORING: The cake will keep for about 2 days at room temperature and, actually improves. It's best not to cover it as it's too moist. Leave the cake on its plate and just press a piece of plastic wrap or wax paper against the cut surfaces.