**Prepare Ingredients:**
Add the chicken, rice, carrots, celery, onion, garlic, chicken broth, thyme, parsley, salt, and pepper to the slow cooker. Stir to combine.
**Cook the Soup:**
Cover and cook on low for 6-7 hours or on high for 3-4 hours, until the chicken is fully cooked and tender.
**Shred the Chicken:**
Remove the chicken from the slow cooker, shred it with two forks, and return it to the pot.
**Add Lemon Juice:**
Stir in the lemon juice. Taste and adjust the seasoning as needed.
**Optional Thickening Step:**
Whisk the eggs and additional lemon juice in a small bowl. Gradually ladle a small amount of the hot broth into the egg mixture, whisking constantly to temper the eggs. Slowly pour the mixture back into the soup while stirring continuously for a creamy texture.
**Serve and Garnish:**
Ladle the soup into bowls, garnish with fresh parsley and lemon slices, and serve warm.