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 and add the dried oregano, salt, and black pepper.
Cover and cook on low for 4 hours, or until the chicken is cooked through and vegetables are tender.
Remove the chicken breasts from the slow cooker and transfer to a cutting board. Shred the chicken using two forks.
Add the rinsed rice to the slow cooker and stir to combine.
Return the shredded chicken to the slow cooker. Cover and cook on high for an additional 30 minutes, or until the rice is tender.
In a medium bowl, whisk together the lemon juice, eggs, and olive oil until smooth.
Temper the egg-lemon mixture by slowly ladling about 1 cup of hot soup broth into the bowl while whisking continuously.
Gradually pour the tempered egg-lemon mixture back into the slow cooker, stirring constantly to combine. This will thicken and enrich the soup without curdling the eggs.
Stir in the chopped fresh parsley.
Taste and adjust seasoning with additional salt, pepper, or lemon juice if desired.
Serve the soup hot, garnished with extra parsley if preferred.