Classic Chicken Pot Pie

Name: Classic Chicken Pot Pie
Cook time: 45 min
Total time: 45 min
Cuisine: American
Ingredients: 1 recipe Flaky All-Butter Pie Crust; 1/3 cup butter; 1 large yellow onion, finely diced; 3 stalks celery, finely diced; 3 small-medium carrots, finely diced; 4 cloves garlic, minced; 1/2 teaspoon dried thyme; 1/3 cup all-purpose flour; 1 3/4 cups chicken broth; 2/3 cup half-and-half, plus an additional splash; 3 cups cubed cooked chicken (store-bought rotisserie is fine!); 1 cup frozen peas, not thawed; 1 large egg; Kosher salt and black pepper