Dice the carrot into ¼-inch pieces, chop the celery and onion into similar-sized pieces, and mince the garlic. Heat the olive oil in a large pot over medium heat until shimmering, then add the carrots, celery, and onion.
Sauté for 5-8 minutes, stirring occasionally, until the vegetables soften and begin to release their aromatics.
Add the minced garlic and stir constantly for about 30 seconds until fragrant. Pour in the oregano and parsley, stirring to coat the vegetables evenly. Immediately pour in the chicken broth and increase the heat to bring the liquid to a boil.
Once the broth is boiling, stir in the rice and return to a boil. Cover the pot with a lid and reduce the heat to low, simmering for about 20 minutes until the rice is tender.
While the rice cooks, whisk the room-temperature eggs together with the lemon juice in a small bowl until well combined.
Once the rice is tender, remove the pot from heat and let it cool for about 1 minute. Slowly ladle one ladle of the hot broth into the egg-lemon mixture while whisking constantly.
Slowly pour the tempered egg-lemon mixture back into the pot while stirring gently and constantly. Add the shredded chicken, salt, and pepper, stirring to distribute evenly.
Heat gently over low heat for 1-2 minutes just until warmed through, but do not boil.
Ladle the soup into bowls and top each serving with a sprinkle of fresh parsley. Serve immediately with crackers on the side.