Instructions

1. Make the pastry: In a large bowl, stir together the flour, grated cheese, and salt. Add the butter and, using your fingertips to incorporate, gently squeeze and toss the mixture until the butter has mostly broken down, with some pea-size pieces remaining. Drizzle in ¼ cup of ice water, then use a fork to gently combine just until the dough holds together and is smooth with visible flecks of butter. (If the mixture feels too dry, add more water, 1 tablespoon at a time, until a shaggy dough forms.) Shape the dough into a ½-inch-thick disk, wrap tightly in plastic, and refrigerate for at least 1 and up to 48 hours. 
2. Make the apples: Position a rack in the top third of the oven and preheat to 400°F. To a 12-inch cast iron skillet over medium heat, add the butter, sugar, brown sugar, vanilla, and salt and cook, stirring occasionally, until the sugar and butter have melted and the mixture comes to a simmer, about 5 minutes. Arrange the apple wedges in concentric circles, making sure they fit snugly to the edge of the skillet. Cook until the apples begin to soften and the liquid has darkened to a light amber caramel, about 25 minutes. Remove from the heat.
3. Remove the dough from the fridge and set aside until pliable, 5–15 minutes. On a lightly floured work surface, roll the dough out to a 12-inch circle, about ¼-inch thick. Drape the dough round over the skillet, tucking the edges gently between the apple wedges and the sides of the pan. Bake until the crust is deep golden brown all over, 20–25 minutes.
4. Transfer the skillet to a wire rack and set aside to cool slightly, about 10 minutes. Then, while the tarte is still warm, cover the skillet with a large inverted plate and, quickly and confidently, flip the tarte out onto the plate. (If any apples stick, use a fork to dislodge and tuck them back onto the tarte.) Set aside to cool for 5 ­minutes more. Garnish with shaved cheese and ­microgreens and serve warm.