Melt the butter in a saucepan over medium heat. Stir in the onion then cook until the onion has softened and turned translucent, about 5 minutes. Place the chicken breasts into the saucepan skin-side-down. Pour in the chicken stock and enough water to just cover the breasts. Bring to a simmer over high heat then reduce heat to medium-low, cover then simmer until the chicken begins to fall off the bone, about 1 hour. Remove the chicken breasts from the saucepan. Remove and discard the skin and bone; allow the meat to cool then tear apart into bite sized chunks.
While the chicken breasts 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 butter in with your fingers until the mixture resembles coarse sand. Make a well in the centre and pour in the milk. Stir together until a stiff dough forms. Work the dumpling dough adding a bit more milk if needed to just make the dough come together. Turn out onto a floured surface then knead until smooth. Roll the dough out to about a 1cm thickness. Using a pizza cutter, cut the dumplings into squares.
After you've shredded the chicken, stir the cream of chicken soup, 2 cups water, salt, pepper and shredded chicken back into the saucepan. Return to a simmer over medium-high heat. Drop the dumplings into the simmering soup one at a time to keep them from sticking together. Cover, reduce heat to medium-low then simmer until the dumplings have puffed and are no longer doughy in the centre, 10 to 20 minutes.
Use bone in chicken to make a good stock. If you want to use chicken without bones remember to add a chicken stock cube.