Combine diced peaches, granulated sugar, brown sugar, cornstarch, lemon juice, cinnamon, nutmeg, and salt in a medium saucepan.
Cook over medium heat, stirring occasionally, for 6 to 8 minutes until peaches soften and juices thicken. Stir in butter until melted.
Remove from heat and let the filling cool for about 10 minutes until warm, not hot.
Lightly flour a clean work surface, unfold one pastry sheet, and gently roll to smooth fold lines. Cut into 4 equal squares. Repeat with second sheet.
Spoon about 2 tablespoons of cooled filling into the center of each square, leaving a clean 1/2-inch border.
Fold each square diagonally into a triangle, press edges to seal, and crimp with a fork. Transfer to a parchment-lined baking sheet.
Refrigerate the shaped turnovers for 10 minutes while the oven preheats to 400°F (204°C).
Whisk egg and water. Brush a thin layer of egg wash over each turnover, avoiding cut edges, and sprinkle with turbinado sugar.
Cut two small slits in the top of each turnover to let steam escape.
Bake for 18 to 20 minutes until deep golden brown and puffed. Rotate the tray halfway if needed.
Transfer to a wire rack and let cool for at least 10 minutes before glazing or serving.
Whisk powdered sugar, milk, and vanilla together until smooth. Drizzle over cooled turnovers and let set for 5 minutes.