Preheat the oven to 350°F (180°C). Grease the inside edges of a 9-inch (23-cm) round cake pan with butter.
Pour the melted butter into the pan and tilt to coat the bottom. Sprinkle with brown sugar and drizzle with maple syrup.
Wash, core, and peel the apples, then cut them in half from top to bottom. Place the halves flat side down on a cutting board and slice into thin pieces about ⅛ inch (3 mm) thick. Arrange the slices in a circular pattern over the sugar mixture in the pan.
In a mixing bowl, beat the granulated sugar and softened butter with a mixer at medium-high speed until pale and fluffy, about 1 to 2 minutes. Add the eggs and vanilla extract and mix until smooth.
Sprinkle half of the flour, along with the cinnamon, baking powder, and salt, into the bowl. Mix on low speed until just combined.
Add the milk and mix on low until blended.
Add the remaining flour and switch to a spatula or wooden spoon. Fold the batter gently until the flour is just incorporated. A few streaks of flour are fine.
Spoon the batter over the apples and gently smooth the top. Bake for 35 to 38 minutes, or until golden brown. The cake should spring back when pressed lightly in the center, and the edges will begin to pull away from the sides of the pan.
Transfer the pan to a wire rack and let it rest for 10 minutes. Place a serving plate on top, then carefully flip the cake over to release it from the pan. Serve warm or at room temperature.