Step 1

Arrange a rack in the middle of the oven and heat to 350°F. Lightly coat a 9x13-inch baking pan with baking spray or olive oil.

Step 2

Place the yogurt, granulated sugar, oil, lemon juice, eggs, and vanilla in a large bowl and whisk to combine. Peel and core the apples, then chop into rough 1/2-inch chunks. (Do not grate the apples or the batter will be too wet.) You should end up with 4 to 4 1/2 cups of apples. Add the apples to the yogurt mixture and stir to combine.

Step 3

Add the flour, baking powder, baking soda, salt, nutmeg, and 1/2 teaspoon of the cinnamon. Stir just until no dry pockets of flour remain; set aside. Place the remaining 2 teaspoons cinnamon, brown sugar, and butter in a small bowl and smash with a rubber spatula to combine.

Step 4

Pour 1/2 of the batter into the pan. Sprinkle with 1/2 of the cinnamon-brown sugar mixture on top, dropping it on the batter in small lumps. Spread the rest of the batter over the top, then sprinkle with the remaining cinnamon-brown sugar.

Step 5

Bake until a tester inserted in the center comes out clean, covering with aluminum foil toward the end if the top is browning too much, 45 to 55 minutes. Place the pan on a wire rack and let it cool for at least 1 hour. Cut and serve the cake warm or at room temperature.