To prepare crust, lightly spoon 1 cup flour into dry measuring cups; level with a knife. Combine 1/4 cup flour, ice water, and vinegar in a small bowl. Combine 3/4 cup flour and 1/4 teaspoon salt in a large bowl; cut in shortening with a pastry blender or 2 knives until mixture resembles coarse meal. Add vinegar mixture; stir just until moist. Press mixture gently into a 5-inch circle on heavy-duty plastic wrap; cover with additional plastic wrap. Chill for 15 minutes. Roll dough, still covered, into a 13 x 10-inch oval. Place dough in freezer 5 minutes or until plastic wrap can be easily removed.
Preheat oven to 400°.
To prepare filling, bring 2 1/2 cups Chicken Stock to a boil in a medium saucepan. Add potato and carrot; cook 2 minutes. Drain mixture in a colander over a bowl, reserving cooking liquid.
Melt butter in a large nonstick skillet over medium heat. Add shallots; cook 3 minutes. Lightly spoon 1/2 cup flour into a dry measuring cup; level with a knife. Combine 1/2 cup flour and 1/2 cup Chicken Stock; stir with a whisk. Add to skillet. Stir in potato mixture, reserved cooking liquid, chicken, peas, 3/4 teaspoon salt, thyme, and pepper. Cook 10 minutes. Remove from heat; cool slightly. Spoon chicken mixture into a 1 1/2-quart casserole dish coated with cooking spray. Remove 1 sheet of plastic wrap from dough. Place dough on top of chicken mixture, pressing to edge of dish. Remove top sheet of plastic wrap. Cut 5 slits in top of crust to allow steam to escape. Gently brush crust with milk. Bake at 400° for 45 minutes or until golden. Let stand 10 minutes.