Sear chicken: Heat 2 tablespoons oil over medium-high heat in a large Dutch oven/soup pot. Season chicken with ½ teaspoon salt and ½ teaspoon pepper then sear until golden, about 2 minutes per side. Remove to a plate but leave drippings.
Saute vegetables: Melt 2 tablespoons butter over medium-low heat in the drippings. Increase to medium-high heat and add onions, carrots and celery; sauté until tender, scraping up the golden bits on the bottom of the pan. Add garlic and red pepper (if using) and sauté for 30 seconds.
Add ingredients: Add chicken back to the pot along with lemon zest, bouillon, all seasonings and chicken broth.
Simmer soup: Cover the soup to bring to a boil, then reduce to a gentle simmer. Simmer the soup (covered) until the chicken is tender enough to shred, about 12-15 minutes (longer for breasts). Meanwhile:
Make avgolemono: While the soup is simmering, whisk together the lemon juice, eggs and egg yolks in a medium liquid measuring cup or mixing bowl; set aside.
Shred chicken: Once chicken is tender, remove to a cutting board; shred once cool enough to handle, don’t add back to the pot yet.
Simmer rice: Add the rice to the soup, cover, and bring to a simmer. Simmer gently until rice is almost tender, about 8-10 minutes. Remove the pan from the heat. ***NOTE: Cooking time may vary depending on your rice and how long it takes to bring your soup to a simmer so I suggest checking the rice often until done. Don't cook until completely tender (should have a little "bite" to it) because the rice will continue to cook while the hot soup sits.
Temper eggs: Using a ladle, slowly drizzle some of the hot soup into the egg and lemon mixture while whisking CONSTANTLY in order to temper the eggs. Repeat about 4 times, whisking constantly.
Add eggs to soup: Off-heat, whisk the tempered egg mixture into the soup, whisking constantly.
Add chicken and herbs: Stir the shredded chicken, fresh parsley and dill into the soup. Season to taste with additional salt and pepper if desired.