Cut a chicken into quarters and simmer in a gallon of water with carrots, onion, and celery. When tender, remove and bone chicken. Return bones to the pot and reduce liquid by about a third. Strain and return to pot to simmer.
You want a good, rich broth.
Make a stiff biscuit dough with sweet milk; roll it out to about an eighth of an inch, cut into strips, and drop into boiling broth. As the liquid thickens, add the chicken meat, boil for another minute, then reduce heat and cover.
After five minutes, cut the heat, and salt to taste. I like chicken and dumplings with a stiff dose of black pepper.