Preheat oven to 350 degrees F. Coat 4 individual size baking dishes with cooking spray.
Season the chicken with 1/4 teaspoon of the salt and 1/4 teaspoon of the pepper. Heat 2 teaspoons olive oil in a large skillet over medium high heat. Add chicken and brown on all sides, about 2 1/2 minutes per side. Transfer to a plate.
Add 2 more teaspoons of oil to skillet. Saute leeks and celery, stirring occasionally, about 3 minutes. Add potatoes, green beans, garlic and remaining 3/4 teaspoon salt and 1/4 teaspoon pepper. Cook 2 minutes more until all ingredients are combined.
Add milk. Mix broth with flour, then add to pan. Cook, stirring, until mixture comes to a boil. Reduce heat to medium low and simmer 10 minutes. Return chicken, add peas, parsley and thyme. Spoon into prepared baking dishes.
Cut phyllo dough into quarters. Place one piece on each pie. Brush with oil. Repeat until there are 3 layers. Tuck ends of dough into dish. Sprinkle with Parmesan cheese.
Place baking dishes on a baking sheet and bake 30 minutes, until filling is hot and bubbling.