Preheat the oven to 350 degrees F. Lightly grease an 8- or 9-inch round pan with cooking spray or butter, then line the pan with a round of parchment paper cut to fit the inside of the pan. Grease the parchment paper with cooking spray or butter.
In a medium bowl, whisk together the flour, baking powder, and salt. Set aside.
In a large bowl, whisk together the butter, eggs, brown sugar, and vanilla until creamy, lighter in color, and slightly foamy, about 30 seconds to 1 minute.
Whisk in the dry ingredients until just combined. Stir until there are no more streaks of flour.
Use a rubber spatula to fold the apples into the batter until evenly coated. (It will look like there isn't much batter, but don't panic!) Scrape the batter into the prepared pan and use your spatula to nudge the apples into a mostly even layer.
Bake until golden brown on top and a cake tester or toothpick inserted into the center comes out with just a few moist crumbs clinging to it, about 50-55 minutes.
Remove from the oven and set on a wire rack to cool for 15 minutes. Carefully run a dull knife around the edge of the cake to free it from the pan, then turn the cake out of the pan onto the wire rack.
Let cool for at least 15 minutes more. (For the neatest pieces, slice with a sharp serrated knife, which will help you saw through the apple peices, rather than crushing them.)
Notes
This cake is best served the day it's baked, but it can be stored, uncovered, at room temperature for up to 3 days. (Press parchment paper or plastic wrap directly onto cut surface, but don't cover the whole cake as it will become damp.)
Use any variety of apple you like, such as Cosmic Crisp, Honeycrisp, or Granny Smith.
Add 1 teaspoon cinnamon to the cake batter for more flavor.
Serve plain or topped with whipped cream or vanilla ice cream. It's also yummy with plain yogurt for breakfast.