Skin and cut up the bird, leaving bones in the dark meat. Save the skin. Fry the skin in a large pot on medium heat, until light brown. Pull out the skin and discard, leaving drippings behind. Sprinkle chicken pieces with paprika, and, working in small batches, fry the pieces until golden brown. Set aside. (You can begin work on step 6 while the chicken is frying.) Lower heat and sweat the onion and salt in the drippings. When onion is tender, add the pepper and paprika and fry for a few minutes more. Add a splash of the water and scrape any fond from the bottom. Add remaining water and stir. Add the chicken pieces and simmer until chicken is cooked through. Bring a large pot of salted water to a boil. Beat the eggs and add salt and milk; whisk to mix. Add 2 c flour and mix with a heavy spoon. Add the third cup and mix. With the fourth cup, you will have to lose the spoon and knead by hand. The dough should be thick and rubbery, so knead well, mixing in all the flour. Divide into 5 or 6 portions. Working in batches, take one of the dough portions and tear off small (1/2") pieces, throwing right into the boiling water. Work quickly. Wait for the water to boil and all the dumplings to float, then cook for 1 minute more. Remove with a slotted spoon into a colander to drain. Repeat with the remaining dough. When the chicken is cooked, remove it from the broth. Whisk in the sour cream. Return to a simmer. When the broth is simmering, whisk the cornstarch with cold water, and add to thicken to your liking. The 2 T gives a medium-thick gravy. Return the dumplings (and, if you prefer, the chicken) to the pot and return to a simmer. Serve the chicken smothered in sauce and dumplings.