Preheat your oven to 375°F (190°C) and grease a 9-inch pie dish or a 9x13-inch baking dish.
In a large skillet, melt butter over medium heat. Add onions and garlic, cooking until fragrant. Stir in carrots, peas, and corn, cooking for another 2-3 minutes.
Sprinkle the flour over the vegetables, stirring continuously for 1 minute. Slowly add chicken broth, stirring until thickened. Stir in milk or heavy cream, seasonings, and cooked chicken. Simmer for 2-3 minutes until creamy and thick.
Pour the chicken filling into the prepared dish and spread it evenly. Roll out the pie crust or puff pastry and place it over the filling, tucking in the edges.
Brush the top with egg wash and cut small slits in the pastry to allow steam to escape.
Bake for 35-40 minutes, or until the crust is golden brown and the filling is bubbling.
Let the pot pie rest for 5 minutes before slicing. Enjoy with a fresh salad or roasted vegetables.