Preheat the oven to 350°F (175°C) and set an oven rack in the middle position. Grease a 9-inch springform or regular cake pan with butter or nonstick cooking spray. If using a regular cake pan, line the bottom of the pan with parchment paper and grease again.

In a small bowl, whisk together the flour, baking powder and salt.

Using a handheld mixer with beaters or a stand mixer with the paddle attachment, cream the butter and granulated sugar until light and fluffy, about 3 minutes. Add the eggs, one at a time, beating well and scraping down the sides of the bowl after each addition. Beat in the vanilla and rum. Don't worry if the batter looks curdled at this point; that's okay. Add the flour mixture and mix on low speed until just combined. Using a rubber spatula, fold in the chopped apples.

Scrape the batter into the prepared pan and even the top. Sprinkle evenly with 1 tablespoon of granulated sugar. Bake for about 40 minutes, or until the cake is golden and a toothpick inserted into the center comes out clean. Allow the cake to cool on a rack in the pan. Once cool, run a blunt knife around the edges of the cake. If using a springform pan, remove the sides. If using a regular cake pan, carefully invert the cake onto the rack, remove the parchment paper, then gently flip the cake over and place right-side-up on a platter. Using a fine sieve, dust with confectioners' sugar (if using). Cake can be served warm or room temperature, plain or with lightly sweetened whipped cream or vanilla ice cream.