Preparation

• Gather all ingredients.
• 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.
• Whisk flour, baking powder, baking soda, and ½ teaspoon sea salt together in a bowl; set dry ingredients aside.
• 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.
• Beat 1 cup white sugar and ½ cup butter together in a separate bowl with a spatula until well blended.
• Whisk in 1 egg until mixture is smooth, 2 to 3 minutes.
• Whisk in second egg until thoroughly incorporated. Add yogurt and vanilla; whisk together.
• Add flour mixture to yogurt mixture; whisk just until flour disappears. Do not overmix.
• 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.
• Fold apples into cake batter with a spatula until just combined.
• Spread ½ the batter into the prepared baking dish; evenly scatter ½ crumble mixture over top.
• Dollop remaining ½ batter on top crumble mixture; spread carefully to evenly distribute, trying not to disturb the crumbs.
• Top with remaining ½ crumble mixture; press crumbs into the batter very gently.
• Bake in the center of the preheated oven until a toothpick or bamboo skewer inserted into the center comes out clean, about 40 minutes.
• Cool to room temperature before slicing and serving, about 30 minutes.