Place the chicken breasts in the bottom of the slow cooker.
Add the chopped onion, diced carrots, diced celery, and minced garlic on top of the chicken.
Pour in the chicken broth, then add the dried oregano, dried thyme, kosher salt, and ground black pepper.
Cover and cook on low for 4 hours, until the chicken is cooked through and vegetables are tender.
Remove the chicken breasts from the slow cooker and shred them using two forks.
Return the shredded chicken to the slow cooker.
Add the rinsed rice to the slow cooker, stir to combine, then cover and cook on high for 30 minutes, until the rice is tender.
While the rice cooks, in a medium bowl, whisk together the eggs and lemon juice until smooth.
Temper the egg-lemon mixture by slowly adding 1 cup of hot broth from the slow cooker into the bowl while whisking constantly.
Gradually pour the tempered egg-lemon mixture back into the slow cooker, stirring gently to combine.
Cook on low for an additional 5 minutes, stirring occasionally, until the soup thickens slightly. Do not boil after adding the egg mixture to prevent curdling.
Stir in the chopped fresh parsley.
Taste and adjust seasoning with additional salt or lemon juice if desired.
Serve hot.