Place the chicken in a large pot with the pieces of carrot, celery, and onion, as well as the bay leaf, peppercorns, and salt. Add the 2 cups broth and just enough water to cover by about 1 inch. Bring the water to a boil and reduce heat to medium-low. Simmer, partially covered, about 1 hour to 1 hour and 15 minutes, or until the meat is tender. Remove the chicken and set aside on a cutting board until cool enough to handle, then strain the broth. Measure out 3 cups and reserve the rest for another recipe or freeze for soup.
Melt the butter in a large deep oven-proof skillet (or a Dutch oven if that’s what you have) over medium-high heat and add the chopped carrots, celery, onions or leeks, and mushrooms. Cook, stirring frequently, about 8 minutes, until golden. Add the garlic and stir frequently until you can smell the garlic, about 1 minute.
Sprinkle over the flour and cook, stirring frequently for about 2 minutes, until the flour has coated the vegetables and has started to turn golden brown. Slowly add the reserved 3 cups cooking stock, whisking constantly, then whisk in the milk and bring to a simmer.
Add the peas and sage. Season with salt and pepper. Reduce the heat and simmer, stirring occasionally, for about 10 minutes or until thick and creamy.
While the vegetable mixture is simmering, remove the skin from the chicken and discard. Take the meat off the bones, keeping the chunks a generous 1 to 2 inches, and then add the chicken back into the pot with the vegetables.
For the Dumplings: Combine the flour, baking powder, and salt. Whisk in the milk and parsley just until combined.
Drop the dumpling dough on top by the heaping tablespoonful. Cover the pan tightly and simmer over medium heat for about 15 minutes without peeking. Serve immediately.
Notes
You can use 4 pounds of bone-in skin-on chicken pieces instead. If you want to use 3 pounds of boneless, skinless thighs and breasts, you can, and it will cook a bit faster, but you won't get a flavor that is as deep.
Serve immediately, either from the hot pan or transfer the mixture to a shallow serving bowl, trying to keep the biscuits on top.