Set the puff pastry sheets out to thaw until pliable but still cool to the touch. Keep them on parchment or a light dusting of flour so they don’t stick. If your kitchen is warm, pop them back in the fridge between steps.
Make the filling. In a wide skillet over medium heat, melt the butter. Add the diced apples, brown sugar, granulated sugar if using, cinnamon, nutmeg, and salt. Stir to coat and cook 5 to 7 minutes, until the apples soften but still hold shape.
Stir in the lemon juice and vanilla. Sprinkle in the cornstarch and cook 1 to 2 minutes, until the juices look glossy and lightly thickened. Remove from heat. Let the filling cool to room temperature. This matters. Hot filling makes the pastry melt and leak.
Heat the oven to 400°F (200°C). Line 2 baking sheets with parchment paper.
On a lightly floured surface, gently unfold one pastry sheet and roll it into a 10 by 10 inch square. Trim the edges if needed for neatness, then cut into 4 equal squares. Repeat with the second sheet for 8 squares total.
Brush a thin line of egg wash around the edges of each square. Spoon 2 to 3 tablespoons of cooled apple filling into the center of each one. Try not to overfill. I know, it’s tempting.
Fold each square into a triangle, pressing the edges together to seal. Use the tines of a fork to crimp all the way around. Transfer to the prepared baking sheets.
Chill the assembled turnovers in the fridge 10 to 15 minutes. This quick chill helps them puff nicely and keeps the layers defined.
Cut 2 small slits on top of each turnover for steam to escape. Brush the tops with more egg wash and sprinkle generously with coarse sugar.
Bake 20 to 23 minutes, rotating the pans halfway through, until the turnovers are deeply golden and the pastry is puffed and crisp. Visible bubbling at the slits is a good sign.
Cool on the sheet for 5 minutes, then move to a rack. If glazing, whisk together the powdered sugar, milk, and a pinch of cinnamon. Drizzle lightly once the turnovers are warm, not hot.
Serve slightly warm or at room temperature. Good with coffee. Great with a small scoop of vanilla ice cream if you’re feeling it.