Sauté the vegetables: Heat olive oil in a large pot over medium heat. Add diced onion, carrots, and celery. Cook for 5–7 minutes, stirring occasionally, until softened. Add minced garlic and sauté for 30 seconds until fragrant.
Add chicken and broth: Pour in the chicken broth and add the chicken breasts, bay leaf (if using), salt, and pepper. Bring to a boil, then reduce to a simmer. Cook for 10–12 minutes, until the chicken is cooked through (165°F).
Shred the chicken: Remove the chicken using tongs and transfer to a cutting board. Shred with two forks or slice into bite-sized pieces. Set aside.
Cook the rice: Add the rice to the pot. Simmer uncovered for 15–20 minutes, stirring occasionally, until tender. Add extra broth or water if it thickens too much.
Prepare the egg-lemon mixture: In a medium bowl, whisk eggs and lemon juice until frothy. Slowly ladle 1 cup of hot broth into the egg mixture, whisking constantly to temper.
Thicken the soup: Remove the pot from heat and discard the bay leaf. Slowly stir the egg-lemon mixture into the soup. Add shredded chicken back and stir gently.
Adjust and serve: Taste and adjust seasoning with more salt, pepper, or lemon juice. Serve hot, garnished with fresh dill or parsley.