Heat olive oil in a large Dutch oven or heavy pot over medium-high heat.
Add the chopped carrots, celery, and green onions. Sauté briefly until the vegetables begin to soften.
Stir in the chopped garlic and cook for another minute until fragrant.
Pour in the chicken broth and add the bay leaves. Increase the heat to high and bring the mixture to a rolling boil.
Add the rice, season with salt and black pepper, and stir well.
Reduce the heat to medium-low and let the soup simmer for about 20 minutes until the rice becomes tender.
Stir in the shredded cooked chicken.
In a medium bowl, whisk together the lemon juice and eggs to create the base for the Avgolemono sauce.
Slowly add two ladles of hot broth from the pot into the egg mixture while whisking constantly. This step tempers the eggs so they don’t scramble.
Once the mixture is smooth, slowly pour it back into the soup while stirring.
Remove the pot from heat immediately to keep the soup silky and smooth.
Serve the Avgolemono Soup Recipe hot and garnish with fresh parsley if desired