Combine the chicken, water, celery, carrot, onion, salt, and pepper in a large covered pot.
Bring to a boil, then reduce the heat and simmer for 2 1/2 hours.
Remove the chicken to a plate and strain the broth.
Measure the broth and add enough water to make 3 cups of liquid.
Make the Gravy:
Mix together the flour and milk in a small bowl.
Return the broth to the pot and bring to a boil.
Stir in the flour and milk mixture.
Cook until thickened, stirring constantly, and simmer for 3 to 5 minutes.
Return the chicken pieces to the gravy and cover.
Prepare the Biscuit Dumplings:
Cut the Crisco into the self-rising flour until the mixture resembles coarse crumbs.
Stir in the milk to form a dough.
Cook the Dumplings:
Drop spoonfuls of the dough into the gently bubbling gravy.
Cover the pot and cook for 20 to 25 minutes.
Before serving, sprinkle the dish with minced parsley.
Notes / Tips / Wine Advice:
Wine Advice
Pair this hearty dish with a light red wine such as Pinot Noir or a full-bodied white wine like Chardonnay. These wines will complement the rich flavors of the chicken and the creamy dumplings.