Preheat the oven to 425°F.
Pour the cooled coconut filling into the chilled bottom crust.
Spread the filling evenly, ensuring the young coconut meat is well distributed.
Unroll the second crust with the parchment paper.
Place the top crust over the filling.
Trim any excess dough from the edges, if necessary.
Pinch the edges to seal the top crust with the bottom crust.
Using the tines of a fork, poke several holes in the top crust to create steam vents.
Bake the pie in the preheated 425°F oven for 10 minutes.
In a small bowl, whisk together the egg and milk.
Remove the pie from the oven after the initial baking time.
Brush the top crust evenly with the egg wash.
Lower the oven temperature to 350°F.
Return the pie to the oven.
Bake for an additional 30 to 40 minutes, or until the crust is golden brown and fully baked.
Remove the pie from the oven.
Allow the pie to cool until the filling is set.
Serve warm or cold.