1 1/2 cups whole-milk yogurt
2/3 cup olive oil
1 lemon, juiced
1 cup sugar
1 1/2 teaspoons vanilla
4 small tart apples, such as Granny Smith
2 1/2 cups all-purpose flour
2 1/2 teaspoons baking powder
3/4 teaspoon baking soda
1/2 teaspoon salt
Pinch freshly ground nutmeg
2 1/2 teaspoons cinnamon, divided
1/2 cup brown sugar
2 tablespoons butter, softened
Heat the oven to 350°F. Lightly grease a 9×13-inch baking pan with baking spray or olive oil.
Whisk together the yogurt, olive oil, lemon juice, sugar, eggs, and vanilla in a large bowl. Peel and core the apples, and chop into thin pieces about 1/2-inch across. You should end up with 3 1/2 to 4 cups of apples. Stir the chopped apple into the liquid ingredients.
Add the flour, baking powder, baking soda, salt, nutmeg, and 1/2 teaspoon right into the liquids and stir just until no lumps remain. In a small separate bowl, mix the remaining 2 teaspoons cinnamon with the brown sugar and butter.
Pour half of the batter into the cake pan. Sprinkle the batter with half of the cinnamon-brown sugar mixture, dropping it on the batter in small lumps. Spread the rest of the batter over top, and sprinkle with the remaining cinnamon-brown sugar.
Bake for 45 minutes, covering with foil at the end if the top is browning. When a tester comes out clean, transfer the cake to a cooling rack and let it cool for at least 10 minutes before cutting. Serve the cake warm or at room temperature.
This keeps very well for several days, and it gets even moister as it sits, due to the apples.