Using a vegetable peeler, remove strips of zest from 1 lemon. If you have cheesecloth on hand, bundle the lemon zest strips, thyme springs, bay leaf and peppercorns together and secure with kitchen twine or tie in a knot. If you don’t have cheesecloth, skip adding the peppercorns as they will be hard to remove. Set aside.
Juice the lemons, removing seeds and pulp before measuring.
Pour olive oil into a soup pot or Dutch oven. Heat on medium until the oil is shimmering. Add the onion, salt and pepper and sauté until the onion softens, about 5 minutes. Add the garlic and sauté until fragrant, about 1 minute.
Add the chicken broth to the onion and garlic mixture. Increase heat to medium-high and bring to a boil. Add the rice, lemon zest, thyme and bay leaf. Reduce the heat and simmer for 10 to 15 minutes, stirring occasionally, until the rice is tender.
Remove the spice bundle or the lemon strips, thyme and bay leaf from the pot; discard.
In a medium mixing bowl, whisk together the lemon juice, eggs and egg yolks. Using a ladle, slowly drizzle some of the hot chicken broth into the egg and lemon mixture, whisking constantly. Repeat a few more times with more hot broth tempering the eggs while whisking and drizzling.
Remove the pot from the heat. Off-heat, whisk the tempered egg mixture into the soup pot. Add the cooked chicken and stir to combine.
Check the seasoning and add salt and pepper to taste. Add additional lemon juice if needed.
Serve immediately garnished with fresh chopped parsley and coarsely ground black pepper.