1. Make the cake: Preheat oven to 350˚. Generously butter an 8-inch cake pan and line the bottom with a round of parchment paper. Set aside.
2. In a medium bowl, whisk the flour with the baking powder, cinnamon, baking soda and salt; set aside. In a small bowl, combine the milk with the vanilla and set aside.
3. Peel both apples. Coarsely grate one apple into a small bowl and set aside. Stand the remaining apple stem side up and, working from the outside in, cut the apple into 1/16-inch slices. When you get to the core, turn the apple around and cut the second side. Discard the cores. Layer the apple slices in concentric circles on the bottom of the prepared cake pan, then set aside.
4. In the bowl of a stand mixer fitted with the paddle attachment, mix the butter with the brown sugar at medium speed until light and fluffy, about 3 minutes. Add the egg and mix until combined, about 1 minute longer. Add the reserved flour mixture to the butter mixture in three batches, alternating with the reserved milk mixture, until just combined. Using a spatula, fold in the reserved grated apple and any juices left in the bowl.
5. Spoon the batter evenly over the sliced apples, then use a spatula to smooth the top and cover the apples completely with the batter. Bake for 35 minutes or until a tester stick inserted into the center of the cake comes out clean. Remove the cake from the oven and let cool for 5 minutes on a wire rack.
6. While the cake cools, make the caramel: In a small saucepan set over low heat, slowly melt the butter. Add the brown sugar, increase the heat to medium and let the caramel cook, whisking constantly, for 3 minutes. Remove the saucepan from the heat and carefully whisk in 1 tablespoon of the half-and-half. Slowly add the remaining 3 tablespoons of half-and-half. Set the saucepan over medium heat and let the mixture simmer, stirring constantly, until the caramel is smooth, about 1 minute longer.
7. Run a knife around the edge of the cake to loosen and unmold it onto a serving plate. Remove and discard the parchment paper. Drizzle the caramel sauce over the top of the cake, reserving any remaining sauce for another use. Serve immediately.