Using a slightly tart apple, like a Granny Smith or Pink Lady, is key to this frozen-yogurt sundae; sautéing brings out the fruit's sweetness. The crispy, salty cheddar crisps that accompany the sundaes are as good with cocktails as they are with desserts.
Preheat the oven to 350°. In a food processor, combine the flour with 1/4 teaspoon of kosher salt. Coarsely grate the cold butter into the food processor. Pulse until the mixture resembles coarse meal. Add the shredded cheddar cheese and pulse twice. Add the cold water and cider vinegar and pulse just until the dough is evenly moistened. Turn the dough out onto a work surface and knead until it just comes together. Wrap the dough in plastic and refrigerate for at least 20 minutes or until chilled.
Line a baking sheet with parchment paper. On a lightly floured surface, roll out the dough to a 9-inch square and transfer to the baking sheet. Bake for about 40 minutes, until golden. Let cool.
In a large skillet, melt the butter. Add the apples and toss to coat. Add the granulated sugar, light brown sugar, cinnamon and nutmeg and season lightly with salt. Cook over moderately high heat, stirring frequently, until the apples are tender and translucent, about 15 minutes.
Add 1/2 cup of water to the skillet and bring to a boil. Remove from the heat. Transfer half of the apples to a blender or food processor and puree until smooth. Scrape the puree into a bowl and freeze until it is cold, about 30 minutes.
Soften the frozen yogurt slightly and transfer it to a large bowl. Fold in the cold apple puree and freeze until the frozen yogurt is firm, about 30 minutes.
Scoop the frozen yogurt into 8 bowls and top with the sautéed apples. Break the cheddar crust into large shards and serve with the sundaes.