Preheat the oven to 400 degrees F.
Sprinkle the chicken with salt and pepper. Heat 2 tablespoons oil and the butter in a large skillet over medium-high heat. Add the chicken, 1/4 teaspoon salt, 1/4 teaspoon pepper and the nutmeg and saute until cooked through, about 5 minutes.
Add the celery, carrots, garlic and onions and saute until translucent, 3 to 4 minutes. Add the beans and mushrooms and continue to cook until the vegetables have softened, 4 minutes. Sprinkle the flour over the vegetables and cook for 2 minutes. Stir in the chicken broth, bring to a simmer and cook for 2 minutes. Stir in the milk and peas and cook until thick and creamy, 8 to 10 minutes. Remove from the heat. Divide the filling evenly among the 4 baking dishes.
Place a phyllo dough sheet on top of each baking dish and brush the top with oil. Bake until golden brown, about 10 minutes.
Nutrition information per serving: 540 calories, 24 g fat