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Preheat the oven to 350°F (175°C). Line the bottom of an 8-inch springform round pan with parchment paper and grease the sides with baking spray or butter. Set aside.
In a large mixing bowl, beat the eggs and sugar with a hand mixer on high speed for about 3 minutes, until light and fluffy. Add the melted butter, Greek yogurt, and vanilla extract, and beat well. Sift in the baking powder, salt, and flour. Beat on medium-low until just combined. Fold the batter gently with a spatula. Set aside.
Arrange the apple slices, peel side down, in a circular pattern on the bottom of the prepared pan, covering as much surface as possible.
In a small saucepan, combine the water, sugar, and cinnamon (if using). Bring to a boil over medium heat, swirling the pan occasionally until the mixture turns a caramel-like amber color. Pour the caramel over the arranged apple slices. You can pour half of it and save the rest to drizzle on top of the cake once baked.
Gently scoop the batter over the apples, spreading it evenly. Place the pan on a baking sheet to catch any potential spills. Bake for 35-40 minutes, or until a toothpick inserted into the center comes out clean. Allow the cake to cool for at least 5 minutes. Run a knife around the edges, then invert the cake onto a serving plate. Slice and enjoy!