Preheat the oven to 350 degrees F. Spray 4 individual-sized baking dishes with cooking spray.
Season the chicken with 1/4 teaspoon salt and a few turns of pepper. In a large nonstick skillet, heat 2 teaspoons of the oil over a medium-high heat. Add the chicken to the pan and cook for 5 minutes, turning once. Transfer the chicken to a plate.
Add 2 more teaspoons of the oil, the leeks, and the celery to the pan and cook until vegetables begin to soften, about 3 minutes. Add the potatoes, green beans, garlic, and remaining salt and pepper and cook for 2 more minutes.
Add the milk to the pan. Stir the flour into the chicken broth until dissolved and add to the pan. Cook, stirring, until the mixture comes to a boil. Reduce the heat to medium-low. Cover and simmer, stirring occasionally, for 10 minutes. Stir in the reserved chicken, peas, parsley, and thyme. Spoon the mixture into the prepared baking dishes.
Put the remaining 2 teaspoons olive oil into a small bowl. Unroll the phyllo dough and cut it into quarters. Place a quarter sheet on top of each baking dish and brush with olive oil. Repeat with remaining 3 baking dishes. Tuck the edges of the phyllo into the dish rim. Top each pie with Parmesan.
Place on a baking sheet and bake until the filling is bubbling, about 30 minutes.