Prep the pan and oven: Heat the oven to 350°F (175°C). Grease a 9-inch round cake pan (at least 2 inches deep) or a 9-inch springform wrapped with foil on the outside to catch drips.
Make the caramel base: In a small saucepan over medium heat, melt 4 tablespoons butter.
Stir in 3/4 cup brown sugar and a pinch of salt. Cook 1–2 minutes, stirring, until glossy and bubbling. Pour into the prepared pan and spread evenly.
Arrange the apples: Toss apple slices with 1/2 teaspoon cinnamon.
Fan them over the caramel in tight concentric circles, overlapping slightly. Fill gaps with smaller pieces.
Mix dry ingredients: In a medium bowl, whisk flour, baking powder, baking soda, salt, cinnamon, and nutmeg.
Mix wet ingredients: In a large bowl, whisk melted butter, granulated sugar, and brown sugar until combined. Whisk in eggs one at a time, then vanilla, sour cream, and milk until smooth.
Combine batter: Add dry ingredients to wet.
Stir gently with a spatula just until no dry streaks remain. The batter will be thick and silky.
Assemble: Spoon batter over the apples and smooth the top. Tap the pan once on the counter to release air bubbles.
Bake: Bake 38–45 minutes, until the top is golden and a toothpick in the center comes out clean or with a few moist crumbs.
Cool briefly and flip: Let the cake rest 10 minutes on a rack.
Run a thin knife around the edge. Invert a serving plate over the pan and flip swiftly. Lift the pan.
If any apples stick, nudge them back on.
Serve: Let set 10 more minutes so the caramel thickens slightly. Serve warm or at room temperature. Add whipped cream, vanilla ice cream, or a drizzle of cream if you like.