Place the whole chicken in a large stockpot and add the chopped celery, carrots, onion, bay leaf, and 2 quarts of water. Add 2 teaspoons of salt and the black pepper. Bring to a boil over high heat, then reduce to a simmer. Skim off any foam that forms on the surface during the first 10 minutes to keep the broth clear. Simmer gently for about 1 hour, or until the chicken is cooked through and tender.
While the chicken cooks, combine the flour, baking soda, and remaining 1/2 teaspoon salt in a mixing bowl. Add the vegetable shortening and cut it into the flour mixture until it resembles coarse crumbs. Pour in the buttermilk and stir just until the dough comes together. Avoid overmixing to keep the dumplings light.
Using a slotted spoon or tongs, carefully remove the chicken from the pot and set aside to cool. Strain the broth through a fine mesh sieve to remove the vegetables and bay leaf, returning the clear broth to the pot. Adjust seasoning to taste.
Once the chicken is cool enough to handle, remove the skin and bones. Shred the meat into bite-sized pieces and set aside.
Bring the broth back to a gentle simmer over medium-low heat. Drop spoonfuls of the dumpling dough into the simmering broth, making small dumplings about the size of a walnut. Cover the pot and cook for 15-20 minutes without lifting the lid, allowing the dumplings to steam and puff up.
Once the dumplings are cooked through and fluffy, stir the shredded chicken back into the pot. Heat everything through for a few minutes. Serve hot, preferably with a side of crusty bread or steamed greens.