1. Bring first 4 ingredients to a boil in a Dutch oven over medium-high heat. Cover, reduce heat to low, and simmer, stirring occasionally, 5 minutes. Increase heat to medium-high; return to a low boil.
2. Place biscuits on a lightly floured surface. Roll or pat each biscuit to 1/8-inch thickness; cut into 1/2-inch-wide strips.
3. Drop strips, 1 at a time, into boiling broth mixture. Add carrots and celery. Cover, reduce heat to low, and simmer 15 to 20 minutes, stirring occasionally to prevent dumplings from sticking.
One roasted chicken yields about 3 cups of meat.