In a large stockpot, heat the olive oil over medium-high heat until shimmering. Add the onions, celery, and carrots to the pot along with a large pinch of salt. Cook until the onions are beginning to turn translucent about 5 minutes. Add the garlic and saute for 1 more minute. 

Add the stock and the chicken breasts to the pot and bring to a boil. Once boiling, reduce to a simmer and cook for 15-20 minutes or until the chicken is cooked through and reaches an internal temperature of 160°F. If any foam forms on the top of the broth, skim it off and discard. Remove the cooked chicken to a plate to cool. Once cooled shred the chicken into bite-sized pieces. 

Meanwhile, while the chicken is cooking, use a sharp vegetable peeler to remove the zest from the lemon in strips, taking as little of the pith as possible. On a flat surface, stack the strips of zest on top of the sprig of thyme and both bay leaves. Tie tightly in a bundle with kitchen twine. 

Add the herb bundle and rice to the stockpot with 2 teaspoons of coarse kosher salt. Stir to combine, cover, then reduce the heat to low and cook for 16-18 minutes, stirring once or twice. Until the rice is tender. 

While the rice is cooking, combine the eggs and 3 1/2 teaspoons of lemon juice whisking until combined. 

Place a fine-mesh strainer over a large bowl or large measuring cup and pour the egg mixture through. You may have to use the back of the spoon to encourage the mixture through the sieve. Discard any egg bits remaining in the strainer. Place the bowl with the strained egg in it near the stockpot and, while whisking constantly, slowly ladle in about 2 cups of the hot broth.

Now, while constantly stirring, slowly pour the egg mixture back into the stockpot. Cook over medium-low heat, stirring constantly until the soup is thick enough to lightly coat the back of the spoon, about 5 minutes, making sure the soup never reaches a simmer. 

Remove and discard the lemon zest bundle. Add the shredded chicken into the soup and keep over low heat, stirring constantly for another 2 -3 minutes until the chicken is warmed through. 

Turn off the heat, taste, and season with more salt and a generous amount of freshly ground pepper to your liking. Ladle into bowls and top with parsley to serve if desired.