Method

1. Gather all ingredients.

2. Preheat the oven to 350 degrees F (175 degrees C) with an rack in the center position. Generously butter a 9x12-inch baking dish with 2 teaspoons butter.

3. Whisk flour, baking powder, baking soda, and ½ teaspoon sea salt together in a bowl; set dry ingredients aside.

4. Combine walnuts, ⅓ cup brown sugar, ⅓ cup white sugar, 3 tablespoons melted butter, cinnamon, and ¼ teaspoon salt in a bowl until walnuts and sugars are thoroughly coated with butter; set crumble mixture aside.

5. Beat 1 cup white sugar and ½ cup butter together in a separate bowl with a spatula until well blended.

6. Whisk in 1 egg until mixture is smooth, 2 to 3 minutes.

7. Whisk in second egg until thoroughly incorporated. Add yogurt and vanilla; whisk together.

8. Add flour mixture to yogurt mixture; whisk just until flour disappears. Do not overmix.

9. Remove and discard cores from apples. Cut apples into ⅛- to ¼-inch slices. Stack a few slices; cut down the center, lengthwise, then dice, crosswise, into cubes.

10. Fold apples into cake batter with a spatula until just combined.

11. Spread ½ the batter into the prepared baking dish; evenly scatter ½ crumble mixture over top.

12. Dollop remaining ½ batter on top crumble mixture; spread carefully to evenly distribute, trying not to disturb the crumbs.

13. Top with remaining ½ crumble mixture; press crumbs into the batter very gently.

14. Bake in the center of the preheated oven until a toothpick or bamboo skewer inserted into the center comes out clean, about 40 minutes.

15. Cool to room temperature before slicing and serving, about 30 minutes.