Prepare the oven and pan. Preheat the oven to 350°F (175°C). Lightly grease a 9-inch round cake pan with sides at least 2 inches deep, using a little extra butter.
Make the topping. Combine the topping butter and brown sugar in a small saucepan over medium heat. Stir until the butter melts and the mixture begins bubbling. Cook for 1 more minute, stirring constantly, until slightly thickened.
Add the flavorings. Remove the saucepan from the heat. Stir in the cinnamon and vanilla, then pour the topping into the prepared pan. Spread evenly across the bottom.
Arrange the apples. Place the peeled apple slices over the caramel, covering the bottom of the pan. Set aside while you prepare the batter.
Combine the dry ingredients. In a medium bowl, whisk together the flour, baking powder, cinnamon, and salt.
Cream the butter and sugars. In a separate large bowl, beat the softened butter until creamy. Add both sugars and beat until light and fluffy. Beat in the eggs one at a time, followed by the vanilla. Scrape down the bowl as needed.
Finish the batter. With the mixer on low speed, add the dry ingredients in two additions, alternating with the milk. Mix just until combined; avoid overmixing.
Assemble and bake. Gently spoon the batter over the apples and smooth the top. Bake for about 40 minutes, until a toothpick inserted near the center comes out clean or with a few moist crumbs, without wet batter.
Cool and invert. Let the cake cool in the pan for 10–15 minutes. Gently loosen the edges with a thin knife. Place a large serving plate over the pan and, wearing oven mitts, hold both firmly together and carefully invert. Lift off the pan. Replace any apple slices that remain stuck.
Serve. Allow the cake to settle before slicing into 8 portions. Serve warm or at room temperature, with optional vanilla ice cream or whipped cream.
