Heat oven to 375 degrees F.
Lightly oil four 10-ounce ramekins or 1 pie dish.
Melt the butter in a large skillet over medium high heat. Add the onions, garlic, peas, corn and carrots to the pan and cook, stirring occasionally until onions become translucent, about 3 minutes.
Whisk in the flour, salt, thyme and pepper. Slowly whisk in the chicken broth and milk all at once. Stir in turkey and season with salt and pepper, to taste.
Cook and stir until thickened and bubbly.
Divide turkey mixture evenly into four ramekins or use 1 round pie dish.
Place the prepared pie crust over the top of the pie. (If making individual pot pies, cut out circles from the pie crust to fit over eack ramekin).
Cut 4 to 6 vents in the top of the crust.
Brush each crust with the beaten egg.
Place in the oven and bake for 30 minutes or until top is golden.