Preheat oven to 425 degrees.
Boil pheasant/chicken and carrots in lightly salted water 10 minutes or until carrots are near tender. Drain. Mix in the uncooked broccoli.
Sauté mushrooms and onions in butter until onions are tender and mushrooms are lightly browned. Add flour and cook for 1-2 minutes, stirring frequently. Add chicken stock and cream, salt, pepper and thyme. Cook until gravy is thickened. Add salt/pepper to taste if needed.
Place one pie crust into a 9” by 1-1/2” or 2” deep glass pie dish. Press into dish with fingers, making sure the crust is smooth. Cut off any excess.
Place the pheasant/chicken, carrot and broccoli mixture evenly in the pie crust. Pour the gravy evenly over the filling.
Place the top crust over the pie and crimp the bottom/top crust together. Brush egg wash over the crust.
Bake at 425 degrees for 25-35 minutes until top crust is golden brown. Let rest for 10 minutes before serving.