Preheat oven to 400°F and make sure the puff pastry sheet is thawed.
In a large deep skillet, melt butter and oil over medium heat.
Add the onions, carrots, celery and potatoes. Saute until the onions are translucent, about 3-4 minutes.
Stir in the garlic, thyme, parsley, salt and pepper. Cook until the garlic is fragrant, about 1-2 minutes.
Sprinkle the flour all over and stir it together, while cooking for 1-2 minutes.
Slowly add the chicken broth while stirring, then stir in the heavy cream.
Reduce heat to medium-low and cook until potatoes are tender and gravy is thickened, stirring often, about 5-8 minutes.
Stir in the turkey, peas, and corn. Remove from heat and pour into 9x13 or 3 quart baking dish.
Roll the puff pastry into a 11x15 rectangle and lay over the filling. Prick the pastry all over with a fork, about 10 times.
In a small bowl, beat the egg and water together until frothy. Evenly brush the egg wash over the pastry.
Bake for 30-35 minutes until the pastry is deep golden brown and the filling is bubbly.
Let rest for 10 minutes before serving.