Preheat your oven to 425°F (220°C).
In a large bowl, combine the sliced peaches and mangoes.
Add the granulated sugar, cornstarch, ground ginger, ground cinnamon, ground nutmeg, lemon juice and vanilla extract. Stir gently until everything is well mixed. Set the mixture aside.
Use a rolling pin to press down on the pie crust disk a few times to soften it. Roll out the bottom crust into a large circle onto a piece of parchment paper, until it is about ¼ inch thick and 12 inches in diameter for a 9-inch pie dish.
Place the pie dish upside down on the center of the rolled-out dough. Using the parchment paper for support, carefully flip the dough and pie dish over so the dough lines the inside of the dish. You want to have about 1.5 inches of overhang on all sides. Take the overhang and roll it under itself to create a thick edge.
Pour the peach mango pie filling into the pie crust, spreading it out evenly.
Roll out the top crust and cut it into 1-inch wide strips. Lay half of the strips horizontally over the pie. Fold back every other strip halfway and lay one strip perpendicular. Unfold the strips over the perpendicular strip and repeat with the remaining strips to create a lattice pattern.
Trim the excess dough from the strips and press the edges together to seal. Crimp the edges decoratively using your fingers or a fork.
In a small bowl, beat the egg with the water to create the egg wash. Brush the egg wash over the lattice crust and the edges of the pie.
Place the pie on a baking sheet to catch any drips. Bake at 425°F (220°C) for 15 minutes. Reduce the oven temperature to 375°F (190°C) and continue baking for 45 minutes to 1 hour, or until the crust is golden brown and the filling is bubbling. If the pie is browning too quickly, place aluminum foil over the top of the pie to protect the lattice topping and edges from burning.
Remove the pie from the oven and let the pie cool on a wire rack for at least 2 hours before serving. This cooling period allows the filling to set, making slicing easier.
Slice and serve!