In a large saucepan, combine chicken, chicken broth, salt, and pepper. Bring to a boil, reduce heat, cover, and simmer for 30 minutes until chicken is fully cooked.
Remove chicken and let cool. Reserve broth, skim fat, and add milk to equal 2½ cups liquid. Cut chicken into ½-inch pieces.
In the same pan, melt butter over medium heat. Add onion and celery; sauté for 3 minutes.
Stir in flour until blended. Gradually whisk in broth-milk mixture. Cook, stirring constantly, until thickened and bubbly.
Add chicken, vegetables, parsley, and thyme. Pour into a 1½-quart deep casserole dish.
Preheat oven to 400°F (200°C).
Roll pastry slightly larger than the dish, cut slits for vents, and place over filling. Trim edges and flute.
Brush with beaten egg.
Bake for 30 minutes, until crust is golden and filling is bubbling.
Rest 10 minutes before serving.