Preheat oven to 325 degrees F (160 degrees C). Insert the base of a 8-inch springform pan lip-side down. Lightly rub with olive oil, then line with parchment. Set aside.
Whisk All-Purpose Flour (1 cup), Ground Cinnamon (1 1/2 teaspoon), Ground Cardamom (1/2 teaspoon), Ground Cloves (1/4 teaspoon), Baking Powder (1 1/2 teaspoon), Baking Soda (1/2 teaspoon), and Salt (1/2 teaspoon) in a medium bowl.
Whisk Egg (1) with Brown Sugar (1 cup), Extra-Virgin Olive Oil (2/3 cup) and Water (1/3 cup) in a large bowl.
Stir flour mixture into egg mixture. Stir in grated Apple (1 cup) and then fold in chopped Apple (1 cup). Scrape batter into the prepared pan and smooth the top.
Arrange overlapping slices of Apple (1 cup) around the top of the cake. Brush apple slices with Olive Oil (1 tablespoon) and then sprinkle with Brown Sugar (1 tablespoon).
Bake in center of oven until a tester inserted into cake comes out clean, about 45 minutes.
Transfer to a rack to cool. Run a knife between the pan and the outer edge of the cake, then remove ring. Let cool completely, about 1 hour.
Serve with Greek yogurt or whipped cream. Keeps well at room temperature for up to 3 days.