How to Make It
Place potatoes in a small saucepan; add water to cover. Bring to a boil over high. Reduce heat to medium; simmer until just tender, 8 to 10 minutes. Drain.
Heat 1 tablespoon of the oil in a large skillet over high. Add beef; cook, stirring occasionally, until browned on all sides, 5 to 6 minutes. Transfer to a plate; sprinkle with 1/2 teaspoon of the salt and 1/2 teaspoon of the pepper.
Add sliced mushroom caps and remaining 1 tablespoon oil to skillet; cook over high, stirring often, until lightly browned, 4 to 5 minutes. Add carrots and thyme. Reduce heat to medium-high; cook, stirring often, until mushrooms have released their liquid, 3 to 4 minutes.
Whisk together flour and 1 cup of the stock in a small bowl until smooth. Add remaining 3 cups stock to mushroom mixture in skillet; bring to a boil over medium-high, stirring occasionally. Stir in flour-stock mixture. Reduce heat to medium; simmer, stirring often, until carrots are just tender and sauce has thickened, 6 to 8 minutes. Transfer mixture to a large heatproof bowl; stir in potatoes, beef, peas, and remaining 1 1/2 teaspoons salt and 1/4 teaspoon pepper.
Preheat oven to 400°F. Coat an 8-inch square glass or ceramic baking dish with cooking spray. Roll 1 of the piecrusts into an 11-inch circle; fit inside prepared baking dish, pressing gently into corners and allowing overhang on sides of dish. Spoon beef mixture over piecrust in dish; top with remaining piecrust. Press together top and bottom piecrust edges to seal, incorporating any piecrust overhang into pressed edges. Cut 5 to 6 slits on top of piecrust in a pinwheel pattern. Whisk together egg and the 2 tablespoons water in a small bowl; brush onto piecrust.
Place baking dish on a rimmed baking sheet. Bake in preheated oven until filling is bubbling and top is lightly browned, 30 to 35 minutes. Let cool 15 minutes; serve.