Bring first 5 ingredients to a boil in a Dutch oven; cover, reduce heat, and simmer 45 minutes or until chicken is tender. Remove chicken, reserving broth in Dutch oven. Cool chicken and broth.
Skim fat from broth. Skin, bone, and cut chicken into bite-size pieces; set aside. Whisk soup and minced onion into broth; bring to a boil.
Stir together biscuit mix, potato flakes, and milk. Turn dough out onto a heavily floured surface; roll or pat dough to 1/4-inch thickness. Cut into 1/2-inch strips. Pinch off 1/2-inch pieces, and drop, 1 at a time, into boiling broth; add chicken. Cover, reduce heat, and simmer, stirring occasionally, 15 minutes.
Quick Tip: Substitute 8 cups chicken broth for water; reduce salt to 1/4 teaspoon. Proceed with recipe using 3 to 4 cups chopped cooked chicken.