Cook the apples, brown sugar, cinnamon, nutmeg, lemon juice, and vanilla in a skillet over medium heat for 4 to 6 minutes, stirring occasionally, until the apples begin to soften.
2 medium Honeycrisp or Granny Smith apples, 1 teaspoon ground cinnamon, ⅛ teaspoon ground nutmeg, 1 tablespoon lemon juice, 1 teaspoon vanilla extract, ⅓ cup packed light brown sugar
Stir together the cornstarch and water, then add the slurry to the apples. Cook for 1 minute more, stirring constantly, until the filling thickens. Transfer to a bowl and cool completely.
1 tablespoon cornstarch, 1 tablespoon water
Preheat the oven to 400°F. Line a baking sheet with parchment paper.
Whisk together the egg and 1 tablespoon of water to make an egg wash.
1 large egg, 1 tablespoon water
Unfold the puff pastry sheets and cut each sheet into four equal squares.
1 package
Spoon about 2 tablespoons of the cooled apple filling into the center of each square.
Brush the edges with egg wash, fold the pastry diagonally into triangles, and press the edges together with a fork to seal.
Cut two or three small slits in the top of each turnover to allow steam to escape.
Brush the tops with the remaining egg wash and sprinkle with coarse sugar, if using.
Coarse sanding sugar
Bake for 20 to 24 minutes, or until the pastry is puffed and deep golden brown.
Cool for 10 minutes.
Whisk together the powdered sugar, milk, and vanilla until smooth, then drizzle over the warm turnovers.
1 cup powdered sugar, 1 teaspoon vanilla extract, 2 tablespoons milk or cream