Preheat the oven to 350°F (175°C). Grease a 9-inch round cake pan with butter and line the bottom with parchment paper. This ensures the cake doesn’t stick to the pan, making it easier to remove once baked, and helps achieve a smooth, clean finish when serving. The parchment paper also prevents the cake from sticking to the bottom, especially when it's delicate or moist.
Peel (or leave the skin on) and core the apples, then cut them into cubes, about 1/4 to 3/8-inch in size. If the cubes are slightly smaller, that's fine, but avoid making them larger as they may not soften properly during baking. Transfer the apple cubes to a large bowl and toss them with apple cider vinegar. The vinegar helps prevent the apples from browning and adds a slight tang to balance the sweetness of the cake.
In a small bowl, combine the flour, baking powder, and salt. This ensures the dry ingredients are evenly distributed, which helps the cake rise properly and maintains a balanced flavor throughout the batter.
In another bowl, beat the butter, 1/2 cup of white sugar, and brown sugar with an electric mixer until the mixture is light and fluffy. Add the first egg and beat on high speed until fully incorporated and smooth. Repeat with the second egg. Then, mix in the rum, vanilla extract, and half of the flour mixture. Once the flour is fully blended in, add the remaining flour mixture and gently fold it in with a spatula. This method ensures a smooth, well-combined batter with a light texture, while the folding process helps maintain its airiness.
Fold the apples into the batter until they are evenly distributed. Transfer the batter into the prepared pan, pressing it in firmly to ensure an even layer. Smooth the top with a spatula and sprinkle with the remaining 3 tablespoons of sugar. This step helps the cake bake evenly, and the sugar on top creates a beautiful, slightly crisp finish as it caramelizes during baking.
Bake in the preheated oven for about 45 minutes, or until a skewer inserted into the center of the cake comes out clean. This indicates the cake is fully cooked and has set properly, with no raw batter remaining inside. Keep an eye on the cake to ensure it doesn't overbake and dry out.
Remove the cake from the oven and allow it to cool in the pan for about 30 minutes. Once cooled, carefully invert the cake onto a wire rack, then place a serving plate on top and flip it back over. This method ensures the cake keeps its shape. Dust the top with powdered sugar for a finishing touch before serving. This adds a light sweetness and a decorative touch to the cake.