To many people, chicken pie with vegetables is a travesty, but this pie is so good that even purists will love it.
Place chicken, bay leaf, salt, and 2 cups water in a 4-quart Dutch oven. Bring to a boil, then reduce heat to low, cover, and simmer for 1 hour or until chicken is tender. Strain broth, discard bay leaf, and return to Dutch oven. Cool chicken, remove meat, and cut into large chunks; set aside. Add carrots and onions to broth and cook, covered, until tender. Drain broth, add enough water to make 2-½ cups liquid, and return to Dutch oven. Stir in peas, mushrooms, seasonings, and chicken. Combine flour and milk and mix until smooth. Stir flour mixture into broth and cook over medium heat, stirring constantly, until mixture boils and thickens. Pour hot mixture into 2-quart casserole.
Roll out pastry to fit top of casserole. Cut slits in top. Place crust over chicken mixture and trim edge, leaving enough to form a ridge. Flute edge and brush with egg wash. Bake at 400 degrees F for 30 to 35 minutes, until crust is golden and filling is bubbly.