Step 1

Place a rack in middle of oven and preheat to 350°. Spread out walnuts on a rimmed baking sheet and toast, tossing once, until golden brown, 10–12 minutes. Let cool.

Step 2

Meanwhile, heat a 10" ovenproof skillet, preferably cast iron, over medium. Add 2 Tbsp. butter and swirl to coat; arrange apple halves cut sides down in a single layer in skillet. Cook apples, undisturbed, rotating skillet on burner as needed for even browning, until cut sides are golden brown, 5–10 minutes (the timing depends on the juiciness of the apples; juicier apples will take longer). Turn apples over and cook on rounded sides just until they start to release their juices and the tip of a knife slides through with slight resistance, about 5 minutes. Transfer apples to a plate, arrange cut sides up, and let cool.

Step 3

Add ¼ cup brown sugar and 1 Tbsp. water to skillet and set over medium heat. Stir with a wooden spoon or heatproof rubber spatula until sugar is dissolved, then cook, swirling skillet occasionally (do not stir at this point), until thick and bubbling in a thin layer, about 1 minute. Let caramel cool in skillet.

Step 4

Meanwhile, pulse toasted walnuts, flour, salt, baking soda, baking powder, nutmeg, and 1½ tsp. cinnamon in a food processor until combined and nuts are finely ground. Transfer walnut mixture to a medium bowl.

Step 5

Combine granulated sugar, remaining ¼ cup brown sugar, and remaining 8 Tbsp. butter in food processor (no need to wash it) and process in long pulses until mixture is light and smooth. Add eggs and vanilla and process in long pulses, scraping down sides of processor once, until mixture is smooth. Add half of walnut mixture and pulse to combine, then add yogurt and pulse just until incorporated. Add remaining walnut mixture and pulse just to combine.

Step 6

Arrange apple halves cut sides down over cooled caramel in skillet, spacing evenly. Fold batter a few times with spatula, scraping sides to make sure everything is well mixed. Scrape batter over apples and work into spaces around apples. Smooth surface (it’s okay if there’s only a thin layer of batter in places; it will rise in the oven).

Step 7

Bake cake until it is browned across the entire surface and the center springs back when gently pressed, 30–40 minutes. Let cake cool in skillet 10 minutes, then run an offset spatula or a small knife around the sides of the pan to loosen. Set a wire rack upside down on skillet and flip over to release cake; carefully remove skillet. If any apples or cake stick to skillet, scrape them off and press back onto the top of the cake.

Step 8

Serve cake warm or room temperature. Slice into wedges and top with dollops of whipped cream and a dusting of cinnamon.

Step 9

Do Ahead: Cake can be baked 2 days ahead. Let cool completely. Store tightly wrapped at room temperature.