Add the butter and cheese to the bowl of a food processor; cover and process until well-combined, about 20 seconds. Add the egg yolk, sugar, and salt; cover and process until combined, scraping bowl as necessary. Add the flour and pecans; cover and process until a dough forms, about 30 seconds more. (Or, for an electric mixer, place butter and cheese in a large bowl and beat with the mixer on medium speed until well-combined. Add the egg yolk, sugar, and salt; beat with the mixer until combined. Finely chop the pecans. Add to beaten mixture with the flour. Beat until crumbly. Use your hands to gently knead mixture until it a dough forms.)
On a cold, clean surface (flour surface, if necessary), use your hands to form dough into a 16x1 1/2x3/4-inch rectangle. Wrap in plastic wrap and place on a baking sheet. Freeze for 1 hour.
Preheat oven to 375°F. Line 2 baking sheets with parchment paper. Remove dough from the freezer to a cutting board. Using a sharp knife, cut the dough into 1/4-inch-thick slices. Place the slices one inch apart on the prepared baking sheets.
In a small bowl, whisk the reserved egg white with 1 teaspoon water. Brush lightly on the dough slices. Bake on separate racks in the preheated oven for 15 minutes or until lightly golden brown, switching racks halfway through baking. Remove and cool completely on the baking sheets on wire racks. To store, place in airtight containers and store at room temperature for 1 day or in the refrigerator up to 3 days. Allow to come to room temperature before serving.
Preheat oven to 400°F. Coat a 3-quart rectangular baking dish with cooking spray; set aside. In a very large bowl mix together the apples, sugar, lemon juice, cinnamon, and salt. Transfer apple mixture to prepared baking dish. Pour apple juice over apple mixture.
Bake, uncovered, about 50 minutes or until the apples are very tender when pierced with a fork, stirring once. Remove from oven; lower oven temperature to 375°F. Carefully mash the apple mixture with a potato masher. Return to oven. Bake, uncovered, for 45 to 60 minutes more or until thickened, stirring occasionally. Remove from oven. Mash again with the potato masher until nearly smooth. (For a very smooth texture, cool slightly and puree in a food processor.) Cool completely. Serve at room temperature or cover and chill up to 1 week.