Heat oven to 350°F. Coat a 3-quart baking dish with nonstick cooking spray.
Heat 2 teaspoons of the oil in a large nonstick skillet over medium-high heat. Season chicken with 1/4 teaspoon of the salt and 1/4 teaspoon of the pepper. Cook 8 minutes, turning halfway through cooking time. Transfer to a plate.
Add remaining oil, onion and celery; cook 3 minutes. Stir in garlic and flour and cook, stirring constantly, 1 minute. Gradually stir in broth and milk. Cook, stirring, until mixture comes to a boil.
Add potatoes, carrots, green beans, remaining salt and pepper. Cover; cook on medium-low, stirring occasionally, 10 minutes. Stir in chicken, parsley and sage. Spoon into prepared baking dish.
Unroll phyllo. Place 1 sheet on top of chicken mixture; lightly coat with cooking spray. Sprinkle with 1 tablespoon of the cheese. Repeat layering. Tuck edges of phyllo inside rim of baking dish.
Place baking dish on a baking sheet; bake at 350°F for 35 to 40 minutes or until filling is bubbly. Let stand 10 minutes.