Heat 1 teaspoon oil in a Dutch oven over medium-high heat. Add celery, carrot, and onion, and sauté 8 minutes or until tender. Add garlic; sauté 2 minutes. Remove from pan, and set aside.
. Combine 3 tablespoons flour, pepper, poultry seasoning, and salt in a zip-top plastic bag. Add chicken; seal bag, and toss to coat. Heat remaining 2 teaspoons oil in pan; add chicken, and cook 7 minutes, browning on all sides. Add broth to pan, scraping pan to loosen browned bits. Stir in reserved vegetables and peas, and cook 3 minutes or until mixture is thoroughly heated.
. Combine milk and 1 tablespoon flour, stirring with a whisk. Add to chicken mixture. Bring to a boil over medium-high heat; reduce heat, and simmer 8 minutes or until slightly thick, stirring constantly. Stir in parsley. Serve over waffles.