Add the chicken to the water in a stockpot and bring to a boil; simmer, covered, about 20 minutes or until tender. Remove the chicken to a plate to cool. Add the onion, garlic, celery, and carrots to the stock and bring to a simmer. Cut or tear the chicken into 2-inch pieces and return it to the stock. Simmer for about 30 minutes, or until all the vegetables are very tender. Stir the cornstarch mixture into the stock if you like a thick sauce. Dumplings: While the vegetables are cooking, bring to a boil 1-1/4 cups water in a small pan. Turn off the heat and add three sprigs of rosemary. Cover and let steep for 30 minutes. Strain the rosemary tea and mix with about 2 cups of biscuit mix, slightly more if mixture is very gooey. Bring the chicken stock back to a low boil and drop scant teaspoons of dumpling batter into the bubbles. Gently push each dumpling aside with a spoon to keep dumplings from clumping together. Add two more rosemary sprigs, lower heat, cover, and simmer 5 to 10 minutes.