This Vegetable Barley Avgolemono is an easy, fresh, and delicious take on traditional chicken soup. It's perfect for weeknight dinners, and it freezes well too! Once you have added the egg mixture, be careful to heat the soup gently and not let it boil--otherwise it could scramble.
Heat the oil in a large pot or Dutch oven over medium heat. Add the onion and carrots and cook until the onions are translucent, about 3 minutes. Push the vegetables to the sides of the pot and arrange the chicken breasts, skin side down, in the middle. Cook, without stirring, until the chicken is well browned on that one side. Sprinkle with 2 teaspoons kosher salt and 1/2 teaspoon freshly ground black pepper. Add 8 cups of water to the pot and bring to a boil. Reduce the heat to very low and barely simmer, partially covered, for at least 1 hour and up to 4 hours. Strain the broth, reserving the chicken and carrots (discard the onion). When the chicken is cool enough to handle, shred into bite-sized pieces. (Soup can be prepared up to this point several days in advance. Refrigerate until ready to proceed.)
Return the broth, chicken, and carrots to the pot and bring to a simmer.
In a medium bowl, whisk the eggs and lemon juice. Whisking constantly, stir about 1 cup of the hot broth into the egg mixture. Then stir the egg mixture back into the pot. Heat VERY gently (do not boil!). Stir in the barley, peas, and dill. Taste, and season with more salt and pepper as needed.