Dice the onion and cooked chicken or turkey. Set aside.
Place the frozen vegetables in a colander and rinse under cool water to remove any ice. Drain well.
In a heavy-bottom pot over medium heat, melt the butter. Whisk in the flour to form a roux.
Add the diced onion, salt, and pepper. Continue whisking and cook for 1–2 minutes, until the onions begin to soften.
Slowly whisk in the milk and broth. Continue whisking until the mixture comes to a low boil and thickens.
Stir in the drained vegetables and diced chicken. Remove from heat and set aside.
Assemble the Pot Pie
If using 9-inch pie pans, this recipe makes enough filling for two pies. If using a 9x13-inch baking dish, use two sheets of puff pastry rolled out and pressed together for the bottom, and two sheets for the top.
Butter the pan well. Place the bottom crust in the pan, pressing it gently into place.
For pie pans, fill each pan halfway with the chicken and vegetable mixture. For a 9x13 dish, pour all of the filling into the pan and spread evenly.
Place the top crust over the filling. Seal the edges by crimping with a fork or pinching the dough together. Cut a small X in the top crust to allow steam to escape.
Baking the Pot Pie
In a small bowl, beat the egg with 1 tablespoon of water. Brush the egg wash evenly over the top crust.
Bake at 425°F for 15 minutes, then reduce the oven temperature to 375°F and continue baking for 20–35 minutes, or until the crust is golden brown and the filling is bubbling.
If the edges begin to brown too quickly, loosely cover them with foil.
Remove from the oven and let cool for 10–15 minutes before slicing and serving.