How to Make It
Preheat oven to 425°F. Melt 2 Tbsp. butter in a 9-inch ovenproof skillet over medium heat. Sauté mushrooms until golden, about 8 minutes. Remove to a bowl.
Melt other 2 Tbsp. butter in skillet. Add onion; sauté 2 minutes. Sprinkle in flour. Cook, whisking, until golden, about 2 minutes. Whisk in broth, then milk. Bring to a simmer. Cook, whisking, until sauce coats back of a spoon, about 2 minutes. Season with salt and pepper.
Stir in mushrooms, vegetables and chicken. Remove from heat. Place crust on top; cut 2 vents in it. Beat egg with 1 Tbsp. water; brush crust. Bake until browned, 25 to 30 minutes.