Heat a 10-inch oven-proof skillet over medium-high heat and add smoked sausage, stirring occasionally until browned, 6 - 9 minutes.
Reduce heat to medium and add onion, carrot, celery and thyme. Continue to cook until the vegetables are soft, about 5 minutes. Mix in the flour then add the broth, Worcestershire sauce and salt. Cook until mixture is thick and bubbly, 5 - 6 minutes.
Turn off heat and work quickly to place the cold pie crust over the pan and fold the excess dough inward. Cut a steam vent in the center of the pie. Brush with the beaten egg and bake for 10 - 12 min until crust is golden brown.