Start with the broth. Toss all the broth ingredients into a large stockpot, cover with at least 2 quarts of water (you can save any extra broth for later) and bring to a strong simmer, about 200 degrees if you're checking. Drop the heat to below a simmer -- look for lots of steaming, and just a couple bubbles on the surface -- and let everything cook 20 minutes. Fish out the pheasant and remove the breast meat. Set it aside in the refrigerator and return the rest of the pheasant to the pot. Cook for as long as it takes for the meat to want to fall off the leg bones, from 45 minutes for a pen-raised bird to 2 1/2 hours for an old rooster.
When the pheasant is done, gently remove it from the broth and let it cool enough to handle. Pick off all the meat from the bones, being sure to remove all those nasty tendons in the pheasant's legs. Put the meat in the bowl with the breast meat. Take the breast meat and shred it, then return everything to the bowl.
Strain the broth. Using a spider skimmer or slotted spoon, remove as many big pieces of vegetable as you can from the broth. Put a fine-meshed strainer that has a paper towel set inside it over a large bowl or pot. Pour the stock through this. You might need to change paper towels halfway through if it gets too gunked up. Pour the broth into a pot and set it on low heat.
To make the stew, heat the butter in a large pot or Dutch oven set over medium-high heat. Saute the carrot, celery and parsnip for 3-4 minutes, stirring often. You don't want the veggies to brown. Add the flour and stir to combine. Everything in the pot will seize up, but that's OK. Drop the heat to medium-low and cook, stirring often, until the flour turns the color of coffee-with-cream. Add the vermouth and stir well, then start adding the broth 1 cup at a time, stirring constantly, until it looks silky. It should take 6-8 cups.
Add the pheasant meat and bring this to a simmer. Cook gently until the veggies are soft, about 30 minutes.
Meanwhile, make the dumpling dough. Mix all the dry ingredients together, then add the melted butter and milk. Stir just to combine -- do not overwork the dough.
Drop the dough by the teaspoonful into the simmering stew. When all the dough is in, cover the pot and cook over low heat for 15 minutes. It is very important that the stew not boil during this time, or your dumplings will get tough.
After 15 minutes are up, uncover the pot and add the peas and parsley, stir gently to combine. Let this cook 1-2 minutes, then turn off the heat. Add salt and black pepper to taste, then the heavy cream. Serve at once.