In a medium saucepan, melt the butter over medium heat. Add the apple and cook for 3 minutes, stirring occasionally. Add the sugar, cinnamon, allspice, nutmeg, and salt. Continue to cook, stirring often, until the apples are softened and caramelized. Let cool to room temperature before proceeding with the recipe (transfer to a bowl and place in the refrigerator to speed this process along).
Line a large baking sheet with parchment paper. Remove the first sheet of puff pastry from the refrigerator. On a lightly floured work surface, roll it into an 11-inch square. With a sharp knife or pizza cutter, slice it into quarters so you have 4 equal squares.
Scoop one-eighth of the apple filling onto one diagonal half of each square (place it in one of the four corners of each square and leave the other half uncovered), leaving a 1/2-inch border at the edges. Brush egg wash lightly on the pastry edges that are nearest to the apples, then fold the uncovered side over the apples, creating a triangle packet. With the tines of a fork, press the two edges where the pastries meat together to seal. Transfer to the prepared baking sheet, leaving 1 inch between each. Repeat with the remaining filling and puff pastry. Transfer to the refrigerator while the oven preheats (or ideally for 20 minutes; refrigerating reduces the risk of the filling leaking out). Keep the eggwash handy.
Place a rack in the center of your oven and preheat to 400°F. Brush the top of each pastry with eggwash, then with a sharp knife, cut two small vents in the top of each to allow steam to escape. Bake the apple turnovers until the pastry is puffed and golden, about 20 to 22 minutes.
While the turnovers bake, prepare the glaze: in a medium bowl, stir together the powdered sugar, milk, and vanilla extract. Adjust the consistency as needed by adding more powdered sugar 1 tablespoon at a time (for a thicker glaze) or more milk 1 teaspoon at a time (for thinner). The glaze should be thick but easy to drizzle. Drizzle over the warm turnovers. Let cool a few minutes, then serve.
