You are starting, of course, with a skinned and cleaned squirrel. I use two for one pot of dumplings. If you have frozen them after cleaning, don't worry about thawing them out before cooking, but do give them a good rinse under cool water before throwing them in the pot. Be on the lookout for fur and other bits that you do not want in your dumplings- trust me, it's worth the time to get them good and clean.
Put the squirrel in a stockpot along with some onion, carrots, and celery and cover it all with cold water. Turn the heat onto low/medium and cover the pot. The goal is to bring the contents to a simmer slowly and then allow it to simmer very lightly for at least a few hours. You do not want a rolling boil, you want to cook the squirrel slowly and gently so you end up with tender, yummy meat.
After an hour or two you are going to see this: scummy stuff floating on the top of your stock. You do not want this in your final product since it likely contains random bits of fur and other things you might have missed when cleaning up the critters. Yes, this is gross. No, it does not mean you should abandon your efforts- just skim this stuff off and throw it away, everything underneath is good and tasty. For real.
After a couple more hours of simmering you will end up with two things: tender, cooked squirrel and broth. The veggies you put in initially have given up all of their goodness and can now be tossed. Take the squirrel and place it on a plate to cool so you can take the meat off of the bone. Either strain the broth through a fine strainer or place it in a bowl for awhile and let the solids sink to the bottom- once that happens you can skim the top one more time and then pour the broth back into your stockpot, stopping before the solids start to join the party.
Take the meat off of the bone and put it, with the broth, back into the stock pot. Add chicken or veggie broth to get your liquid to your desired amount- remember you will need a good amount of broth to be able to cook your dumplings in, but you don't want so much that you mess up the dumpling to meat to broth ratio. You should probably have about 4-6 cups of liquid. Bring the liquid back up to boiling while you make your dumplings.
You have several choices for your dumplings. I made my own, using the whole wheat noodle recipe found here , but you can also use the frozen egg noodles you can buy at any grocery store, or you can use bits of biscuits (either homemade or canned) and drop them into the boiling broth. Whichever method you choose in about 20-30 minutes you will have lovely dumplings. Make a slurry of 2-3 tablespoons of corn starch and 2-3 tablespoons of water and pour it into your broth to thicken things up just a bit and season as desired- some salt and pepper will make it perfect.
Serve on top of mashed potatoes because that is the only acceptable way to eat dumplings. If you are super duper lucky, you will get to watch your children pick at their dinner suspiciously and try to hide bits of squirrel under potatoes, insisting that they did eat it. If you aren't so lucky, everyone will eat happily and you will be a homesteading hero. Go you.