Make the Soup-Pat the chicken dry with paper towels and season generously with salt and pepper. In a large Dutch oven, heat 2 tablespoons of the oil over medium-high heat. Working in two batches, brown the chicken until golden, about 4 minutes per side. If the pot is looking a little dark on the bottom, splash some water and scrape up any browned bits between batches, and keep those juices with the chicken.
Return all the chicken to the pot and add 8 cups water, the bay leaf, thyme, rosemary, and 1 tablespoon salt. Bring to a boil, reduce the heat to a gentle simmer, and cook for 1 hour.
Strain the broth through a colander set over a large bowl (there should be about 8 cups liquid; skim off some of the fat if you want). Discard the bay leaf and herb sprigs. Using two forks or a pair of tongs, pick the chicken meat off the bones (discard the bones and skin and set aside).
In the same Dutch oven, heat the remaining 2 tablespoons oil and the butter over medium-high heat. Add the onion, carrots, and celery and cook, stirring, until the carrots begin to soften, about 6 minutes. Add the flour and cook, stirring, until all the flour has been absorbed and darkens a bit, about 5 minutes. Add the strained broth, the chicken meat, and the potato, bring to a boil, then reduce the heat and simmer until the broth thickens, about 20 minutes. Stir in the cream, if using. Season to taste with salt and pepper.
While the broth simmers, make the dumpling dough-In a small bowl, whisk together the egg, milk, and melted butter. In a large bowl, whisk together the flour, baking powder, and salt. Pour the wet ingredients into the flour mixture and stir gently with a fork until just incorporated (the batter will be thick; don't overmix, which could toughen the proceedings). Spoon 1 tablespoon of the batter and use a second spoon to push it off into the simmering soup. Repeat with the rest of the dough. Cover and cook until the dumplings are fluffy and tender, about 15 minutes.
Divide the soup and dumplings among bowls. Garnish with parsley and enjoy!