Bring the broth to a simmer in a large, heavy saucepan. Add the chicken, reduce the heat to a gentle simmer, and poach until the chicken is firm and just cooked through, 3 to 5 minutes. Remove the chicken and, if the broth is cloudy, strain the liquid through a cheesecloth-lined strainer (this step isn’t critical—it’s just for looks).
Bring the broth back to a simmer in the saucepan, and add the carrots, cooking until they’re just tender, about 7 minutes. Remove them with a slotted spoon and set aside with the chicken.
Meanwhile, bring a pot of salted water to a boil and cook the egg noodles until they’re al dente. Drain them and set aside.
When you’re ready to serve the soup, bring the broth to a boil. Reduce the heat to a simmer. Add the peas, the cooked carrots, and the chicken, and simmer until everything is heated through. Add the cooked noodles, the parsley, and the thyme. Season with salt and pepper. Serve with the Parmigiano, if you like.