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Preheat the oven to 350°F 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 grainy 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. 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.
Freezer Friendly Instructions: The cake can be frozen for up to 3 months. After it is completely cooled, cover it tightly with aluminum foil or freezer wrap. Thaw overnight on the countertop before serving.