Step 1: Prepare the pie crust. In a large bowl, whisk together 2 ½ cups flour and 1 teaspoon salt. Cut in the cold, cubed butter using a pastry blender or your fingertips until the mixture resembles coarse meal with some pea-sized butter chunks. Gradually add ice water, one tablespoon at a time, mixing until the dough just comes together. Divide the dough into two discs, wrap in plastic, and refrigerate for at least 30 minutes.
Step 2: While the dough chills, prepare the filling. Heat olive oil in a large pot or Dutch oven over medium heat. Add chopped onion and carrots, cooking until softened, about 5-7 minutes. Add any optional vegetables like diced potatoes or mushrooms and cook for another 5 minutes.
Step 3: Make the roux. Melt ½ cup butter in the same pot. Whisk in ½ cup flour and cook for 1-2 minutes, stirring constantly, until a light golden roux forms.
Step 4: Slowly whisk in the chicken broth and milk, stirring until the mixture is smooth and thickened. Bring to a gentle simmer.
Step 5: Stir in the shredded cooked chicken, frozen peas, dried thyme, and dried sage. Season with salt and pepper to taste. Remove from heat and set aside.
Step 6: Preheat your oven to 400°F (200°C). On a lightly floured surface, roll out one disc of pie dough into a 12-inch circle. Carefully transfer it to a 9-inch pie plate. Trim any excess dough, leaving about a ½-inch overhang.
Step 7: Pour the chicken filling into the pie crust.
Step 8: Roll out the second disc of dough into an 11-inch circle. Place it over the filling. Trim the top crust to match the bottom overhang. Crimp the edges together to seal the pie. Cut a few slits in the top crust to allow steam to escape.
Step 9: Brush the top of the pie with the beaten egg wash for a golden finish.
Step 10: Place the pie on a baking sheet to catch any drips and bake for 30-35 minutes, or until the crust is golden brown and the filling is bubbly. If the crust starts to brown too quickly, loosely tent with foil.
Step 11: Let the pot pie cool for at least 10-15 minutes before slicing and serving. This allows the filling to set.