Start by making the apple topping. Place the apple cider and granulated sugar in a saucepan set over low heat. Stir until the sugar has dissolved, then increase the heat to high and bring to a boil. Add the vanilla seeds and pod, cinnamon stick, and apple slices, and reduce to a simmer. Simmer for 3 to 4 minutes, or until the apples are just tender. Remove from the heat and set aside to cool.

Once the apples are cool, use a slotted spoon to transfer about ⅓ of the apple slices (with a splash of the syrup) to a blender or food processor, puree until smooth, and set aside. Leave the remaining apple slices in the syrup.

To make the cake, preheat the oven to 350°F (180°C) and place the butter and brown sugar in the bowl of an electric mixer; cream until pale and fluffy. Add the eggs one at a time, beating well after each addition, then beat in the apple puree. Sift together the flour, baking soda, baking powder, and spices, and fold into the batter. Finally, stir in the Greek yogurt until just combined.

Pour the batter into a 9-inch round springform pan lined with parchment paper, and spread until smooth. Starting from the center, lift the remaining apple slices out of the syrup and arrange them in a circle, repeating until the whole top is covered. Bake for 30 to 35 minutes, or until a cake tester comes out clean.

About 10 minutes before the cake is ready, place the remaining apple cider syrup mixture over high heat and bring to a boil. Reduce to a simmer and cook for 10 minutes, until the syrup has reduced and turned a deeper golden color.

When the cake comes out of the oven, drizzle the top with the apple cider syrup and let cool before serving.