1. Heat oven to 350 degrees. Lightly spray an 8-by-1 1/2-inch round cake pan with vegetable spray and set aside. In a large nonstick skillet, melt 1 teaspoon butter over medium-high heat. Add the apples, cut side down, and sprinkle them with 3 tablespoons sugar; cook until the apples are tender and caramelized, 4 to 5 minutes on each side. Remove from heat.

2. Arrange the apples, cut side down, in the prepared cake pan so that they fit snugly together. Fill in any cracks with small pieces of apple, so that the batter wont leak through. Set pan aside.

3. Melt remaining 2 tablespoons butter; set aside. Break egg into a food processor; process for about 5 seconds, then while processor is running, add melted butter and buttermilk through the feed tube.

4. In a small bowl, whisk together flour, remaining 1/2 cup sugar, spices, baking powder, and salt. Add dry ingredients to mixture in food processor. Cover and pulse until batter is combined, about 20 pulses.

5. Pour the batter over sliced apples. Using a spatula, spread batter to completely cover the apples. Bake until cake is golden brown and a tester inserted in the center comes out clean, 40 to 45 minutes. Remove from oven and place on a wire rack to cool for about 15 minutes before inverting cake onto a serving plate.