Pulse 4 Tbsp. butter and biscuit mix together in food processor until crumbly. (Or cut butter into mix with a fork)
Add water one Tbsp. at a time, pulsing after each addition, until dough just starts to come together.
Pour onto a sheet of plastic wrap. Pull the edges of the wrap together and wrap tightly, forming the dough into a ball as you go. Refrigerate 30 min.
Heat 2 Tbsp. of oil in 12" skillet over medium heat. Add mushrooms, shallots, thyme and salt to pan and cook, stirring occasionally, until mushrooms are tender, 8 to 10 min.
Push mushrooms to side and add port. Bring to a boil and cook until port is reduced by half, about 2 min.
Add carrot, vegetable broth and Worcestershire sauce to pan and stir. Bring to a boil, cover and reduce heat to strong simmer. Cook until carrots are tender, about 8 min.
Combine remaining 3 Tbsp. of butter with flour in small saucepan. Cook over medium heat, stirring constantly, until thick and dark brown, about 5 min. Whisk in a little of the hot broth from the mushroom mixture.
Pour flour mixture into mushroom mixture, stirring constantly, and bring to a boil. Continue stirring until sauce thickens, about 2 min. Pour into deep pie dish and set aside to cool.
Preheat oven to 350. Remove dough from fridge and roll out on floured surface. Drape over cooled mushroom mixture. Use a sharp paring knife to trim away any extra dough around the edges; cut a vent in center.
Just before baking, whisk together the egg and water; brush over crust to give the pie crust a nice shine.
Place the pie plate on a rimmed baking sheet. Bake until the crust is a rich brown and the filling is bubbly, about 40 to 45 min.
Remove from oven and cool slightly before slicing. Serve hot or room temperature.