In a deep skillet, melt butter over medium heat. Add onions and sauté until soft and slightly browned, about 5 - 10 minutes.
Stir in garlic and continue to cook until fragrant, approximately 30 seconds.
Deglaze pan with 3 cups of chicken broth. Bring to a boil and simmer on medium-low heat.
In a small bowl, dissolve cornstarch in remaining 1 cup of chicken broth. Slowly pour in the cornstarch slurry into the skillet, stirring constantly to prevent lumps from forming. Continue stirring until the cornstarch is incorporated, approximately 3 minutes.
While simmering add potatoes and carrots. Let simmer approximately 10 minutes.
Gently stir in poultry seasoning, salt and pepper.
Once potatoes are tender (about 5 more minutes), remove from heat. Stir in turkey, green beans and thyme. Let filling cool while the crust is assembled.
Remove dough from the refrigerator and let sit at room temperature until malleable. Roll 1/2 of chilled dough into a 12-inch circle between two pieces of heavy-duty plastic wrap. Remove top layer of plastic; invert and center dough over a 9-inch nonstick deep dish pie pan or into three 6-inch deep dish pie pans. Press crust into place, then remove remaining plastic wrap. Add filling and spread to fill evenly.
Roll the remaining 1/2 of chilled dough into a 10-inch circle between two pieces of heavy-duty plastic wrap. Remove top layer of plastic; invert and center dough over filling. Press dough into place, then trim, seal and flute the edges. Cut four vents in the top dough using a decorative pattern, if desired.
Transfer the pot pie(s) to the freezer and chill at least 30 minutes.
Meanwhile, preheat the oven to 400°F. Remove the pot pie(s) from the freezer and brush the top crust with egg wash, if desired.
Tent the crust with foil and bake until the crust is golden and the filling is beginning to bubble and has thickened, 1 hour 15 minutes - 1 hour 25 minutes. Let cool slightly before serving.