1. Make pastry: In a food processor, pulse flour, salt and pepper. Pulse in butter and shortening until mixture resembles coarse meal. With motor running, add 6 Tbsp. cold water; process just until dough forms a ball, adding more water if necessary. Turn dough out onto a lightly floured countertop; knead until it comes together. Shape into a rectangle, wrap in plastic wrap and chill.
2. Make filling: Place chicken in a saucepan and cover with broth. Bring to a boil over high heat. Reduce heat to medium-low and simmer until chicken is cooked through, 15 to 20 minutes. Using a slotted spoon, remove chicken to a cutting board. Strain broth; reserve. When cool enough to handle, shred chicken into bite-size pieces.
3. Preheat oven to 375ºF. Line a large baking sheet with foil. Mist a 9-by-13-inch baking dish with cooking spray.
4. Melt butter with oil in a large skillet over medium heat. Add potato, onion, carrots, celery and mushrooms, sprinkle with 2 tsp. salt and 1/2 tsp. pepper and stir well. Cover and cook, stirring occasionally, until vegetables have softened, 15 to 20 minutes.
5. Sprinkle flour over vegetables. Cook, stirring, for 3 minutes. Add 2 cups of reserved broth and milk, increase heat to high and bring to a boil, stirring constantly. Reduce heat to medium and cook, stirring, for 3 minutes. Stir in peas, reserved chicken and thyme. If mixture is very thick, thin with some of reserved broth. Season with additional salt and pepper, if desired. Transfer to baking dish.
6. Brush some of beaten egg around edges of baking dish. Roll out dough between 2 pieces of plastic wrap to a 14- by 10-inch rectangle. Roll dough onto rolling pin and unroll it on top of baking dish. Patch any tears. Press pastry to edges of dish, trim if necessary and crimp to seal. Brush top of pie with remaining egg. Cut a few slits in the top to allow steam to escape. Place on baking sheet and bake until top is golden and filling is bubbly, 50 to 60 minutes. Let rest for 10 minutes before serving.