Melt the butter in a large pot over medium heat. Stir in the onion, and cook until the onion has softened and turned translucent, about 5 minutes. Place the chicken breasts into the pot, skin-side-down. Pour in the chicken broth, and enough water to just cover the breasts. Bring to a simmer over high heat, then reduce heat to medium-low, cover, and simmer until the chicken begins to fall off the bone, about 1 hour. Once cooked, remove the chicken breasts from the pot. Remove and discard the skin and bone; allow the meat to cool, then tear apart into bite sized chunks.
While the chicken breast are simmering, prepare the dumpling dough by whisking the flour, baking powder, and 1 1/2 teaspoons of salt together in a mixing bowl. Work the vegetable shortening in with your fingers until the mixture resembles coarse cornmeal. Make a well in the middle and pour in the milk. Stir together until a stiff dough forms. Turn out onto a floured surface, and knead until smooth. Form the dough into 12 pieces, and set aside.
After you've shredded the chicken, stir the cream of chicken soup, 2 cups water, salt, pepper, and shredded chicken back into the pot. Return to a simmer over medium-high heat. Drop the dumplings into the simmering broth one at a time to keep them from sticking together. Cover, reduce heat to medium-low, and simmer until the dumplings have puffed and are no longer doughy in the center, 10 to 20 minutes.
