Preheat the oven to 350°F. Coat 4 individual-size baking dishes with cooking spray.
Season the chicken with 1/4 tsp.of the salt and 1/4 tsp. of the pepper. In a large nonstick skillet, heat 2 tsp. of the oil over medium-high heat. Add the chicken and brown, about 2-1/2 minutes per side. Transfer the chicken to a plate.
Add 2 more tsp. of oil, the leeks, and celery to the pan and cook, stirring a few times, until the vegetables begin to soften, about 3 minutes. Add the potatoes, green beans, garlic, and the remaining 3/4 tsp. salt and 1/4 tsp. pepper and cook for 2 minutes more, stirring to combine everything. Add the milk. Stir the flour into the 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 chicken, peas, parsley, and thyme. Spoon the mixture into the baking dishes.
Put the remaining 1-2/3 Tbs, oil in 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 the oil. Repeat layering with all three layers. Tuck the edges of the phyllo into the dish rim. Sprinkle the top of each pie with the Parmigiano.
Place on a baking sheet and bake until the filling is bubbling, about 30 minutes. Serve hot.