Oh man!  It is delicious!  
		
		
	
	
  And probably as many variations as every other big recipe!  LOL!
It is kind of like a chicken stew, sort of.  Boil the chicken, I use a stick of butter for a whole chicken. Be sure you have enough water to cover the breasts about 2 inches, maybe more.  Depending on how big of a pot you are using.  If you are using a large pot, then fill it at least halfway with the water.  Add your salt and pepper.  I also throw in a little Tony Chachere's seasoning (a cajun spice).  Then after the chicken is well cooked (you really can't overcook) and the broth is condensed down, some, take out about a cup of the broth and mix it with some all purpose flour til you get a lightly sticky dough.  Put it on a floured surface, and roll it to about 1/4 inch thick.  Cut into strips, then into squares.  I use a pizza cutter.  Shred the chicken while the dumplings are drying out a little bit.  You can let them set out for 15 minutes, or all day and just flip them.  The dryer they are, the less they stick.  Drop them into the boiling broth, one by one.  When they float to the surface,they are done.  I drop a little extra flour in there so it will thicken up.  
It is delicious.  Some people also use canned biscuits instead of homemade dumplings, and use a can of either cream of chicken or cream of mushroom for the thickener.  Be bold with the seasoning so it isn't too bland!  You want it to be thick.
Here is a picture of what it should look like:  
http://ourfamilycookbook.com/OtherDinners4.htm  That isn't my webpage, I just tried to find a pic that looked like mine.  The recipe there looks ok, too.  I have served mine to friends that have never seen or even heard of Chicken and Dumplings, and once they get past the "looks" of it, they LOVE it.  It looks bland and boring, but my kids and dh beg for it, and now all their friends do.  I have to make a HUGE vat of it, because if they know I am making it, I have lots of extra "family" that day.
Also, if you "forget" your broth on the stove (I have!) and it cooks down too much, you can always add some more water to it!  Or chicken broth, if you have some!
If you do decide to make it, let me know how it turns out!
Shelly