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Sprinkle thyme and 1/4 teaspoon each salt and freshly ground black pepper evenly over chicken. Lightly coat 12-inch skillet with olive oil spray and heat on medium-high. Add chicken in single layer and cook 3 minutes or until lightly browned, turning pieces over once halfway through cooking; transfer to plate.
To same skillet, add onion and 1/4 cup chicken broth. Cook 5 minutes or until browned, stirring and scraping up browned bits. Add garlic and cook 1 minute, stirring. Stir in parsnips, carrots, and celery, then add remaining 1 1/2 cups chicken broth. Heat to boiling on high. Cover, and reduce heat to maintain simmer. Cook 10 minutes or until vegetables are tender-crisp.
While vegetables cook, place 1 phyllo sheet on work surface; spray lightly with olive oil spray. Top with another sheet. Repeat with remaining phyllo sheets and olive oil spray, lightly coating top sheet with olive oil spray.
In small bowl, stir cornstarch into water to dissolve; stir into vegetable mixture and simmer 2 minutes or until thickened, stirring occasionally. Stir in peas, chopped parsley, reserved chicken, 1/2 teaspoon salt, and 1/4 teaspoon freshly ground black pepper. Heat to simmering, then transfer to 13- by 9- by 2-inch baking dish. Center phyllo on top, tucking in edges if necessary; then cut 5 slits in phyllo.
Bake 15 minutes or until phyllo is golden brown. Garnish with parsley to serve.
Easy as Pie: To prevent pastry from drying out, work with 1 sheet at a time, keeping the others covered.