Place the chicken thighs 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 sprinkle the dried oregano, kosher salt, and black pepper over the ingredients.
Cover and cook on low for 4 hours, until the chicken is tender and cooked through.
Remove the chicken thighs from the slow cooker and place them on a cutting board. Shred the chicken using two forks.
Return the shredded chicken to the slow cooker.
Add the rinsed rice to the slow cooker, stir to combine, and cover. Cook on high for an additional 30 minutes, or 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 ladling about 1 cup of hot soup broth from the slow cooker into the bowl while whisking constantly.
Slowly pour the tempered egg-lemon mixture back into the slow cooker, stirring continuously to combine.
Cook uncovered on low for 5 more minutes, stirring occasionally, until the soup thickens slightly.
Stir in the extra virgin olive oil, chopped fresh dill, and chopped fresh parsley.
Taste and adjust seasoning with additional salt or lemon juice if desired.
Ladle the soup into bowls and serve warm.