Preheat the oven to 350°F Melt the butter in a small saucepan or in a non-metallic bowl set in the microwave. Grease a nonstick 8-inch round cake pan (not springform) with 1 tablespoon of the butter using a pastry brush or paper towel.
Combine the flour, baking powder, and salt in a small mixing bowl. Wash, peel, and core the apples. Cut them in eighths and arrange over the bottom of the pan.
In a medium mixing bowl, whisk the sugar with the eggs until fluffy. Add the flour mixture and whisk until combined. Add the melted butter and whisk again until blended. Pour the batter evenly over the fruit and bake for 40 minutes, until the top is set and golden brown.
Let the cake settle on a cooling rack for 10 minutes. Run a knife around the pan to loosen and flip the cake onto a plate; if any piece of fruit has stuck to the bottom of the pan, scrape it off carefully and place it back on the cake where it belongs. Flip the cake again onto a serving plate - work cautiously so as not to squish the cake between the two plates. Let cool and serve slightly warm or at room temperature.