Don't boil, simmer. In all honesty, the bones of the carcass give a LOT of flavor to the stock, and give a place to anchor the meat so it doesn't get too compact. But, you can try with another chicken later on!!
Put the meat in a good sized pot with a stalk of celery chopped in two, an onion quartered (try to leave a layer of the "Brown" paper skin on for more flavor and color), a carrot cleaned off and chopped in half and either some peppercorns if you have them, or about a Tbs of coarse ground pepper. Put in about a quart of water or two. I don't usually salt my broth. SIMMER, say medium low or low depending on your stove, for a few hours - two minimum, the longer the better. Don't let it get to a boil. After a few hours, remove the meat, and then strain the broth out. If you'd like, you can let it sit in a large jug and skim the fat (don't throw it away!!!! Use it instead of butter for things that use butter - like mashed potatoes! Yum!!).
Wipe the big pot down, and then add chopped up celery, carrots, and onion (without the skin!), and shred and add the chicken. Add the broth back in-but save two cups on the side. Season with pepper and salt and maybe some poultry seasoning. Bring that to a simmer, and cook until the carrots are tender.
In a big bowl, mix together 6 cups of flour, 2 tsp baking powder, 1 tsp poultry seasoning, and 1 tsp salt. Add in 1 cup of some sort of "fat" - butter, or if you have enough (I collect all mine), a cup of the reserved chicken fat that rendered. Mix that in til crumbly. Add two eggs and the two cups of stock, and mix it all together to a dough. Now here, your can go two ways. My husband likes "fluffy" dumplings, so I just take the dough, and spoon glops into the soup, and then cover. It's done when you can stick a fork in the dough and its clean, no stickies.
Other folks like the little chewy dumpling bits, so in that case, you will have to roll your dough out on a floured surface, knead for a bit, and then roll it about 1/4" thick. Use a pizza cutter to cut little 1" dough squares. Take a spatula and dump the little squares into the soup. They are done when they aren't sticky anymore (just fish one out and taste it
That's how my chicken and dumplings usually goes. Whenever I do a big processing, and I'm to the point all the meat is rested, I pretty much always make this out of one of the chickens - the rest go in the freezer. I love it and DH loves it
