Place cranberries in a small bowl. Cover with boiling water. Let soak until plump, about 15 minutes. Drain.
Preheat oven to 350 degrees F (175 degrees C). Grease a baking sheet lightly with cooking spray.
Cut a deep vertical slice into both sides of each apple, being sure not to cut through to the core. Attach 1 apple to a spiralizer fitted with the straight-flat blade and cut into thin half-slices. Repeat with second apple.
Combine cranberries, apple slices, lemon juice, and vanilla extract in a bowl; toss to coat.
Mix white sugar, cornstarch, cinnamon, nutmeg, and salt together in a bowl. Stir into the apple mixture.
Roll pie crust out into a 10-inch round on a floured work surface. Transfer to the baking sheet.
Spoon apple mixture into the center of the crust, leaving a 2-inch border. Lift up edges of crust and fold around apple mixture. Brush edges with egg white. Sprinkle turbinado sugar all over tart.
Bake in the preheated oven until apples are tender and crust is golden brown, 38 to 40 minutes. Transfer to a serving plate and slice.