Combine chicken meat, chicken broth, salt, and pepper in a large saucepan over medium high heat. Bring to a boil, then reduce heat to low. Cover and let simmer for 30 minutes or until chicken is fully cooked.

Remove chicken and let cool. Pour the remaining chicken broth mixture into a measuring cup and skim the fat. Add enough milk to make up 2 ½ cups of liquid. Dice the chicken into ½-inch pieces.

Melt butter in the same pan over medium heat. Add chopped onion and celery, and sauté for 3 minutes. Stir in flour until well blended, then gradually stir in the broth and milk mixture. Simmer while stirring until thickened. Mix in the diced chicken, vegetables, parsley, and thyme.

Preheat the oven to 400°F (200°C).

Roll out the pre-made pastry to fit the casserole. Cut vents in the pastry. Place it over the chicken mixture in the casserole. Trim excess and flute the edges. Optionally, add decorative cutouts with remaining dough. Brush with beaten egg.