In a large Dutch oven or heavy pot, heat the olive oil on medium-high. Add the carrots, celery, and onions, toss them together to saute briefly, then stir in the garlic.
Add the chicken broth and bay leaves then raise the heat to high. Once the liquid has come to a rolling boil, add the rice, chicken, salt, and pepper. Turn the heat to medium-low and simmer for 20 minutes or until the rice is tender.
Prepare the egg-lemon sauce: In a medium bowl, whisk together the lemon juice and eggs. Continue to whisk while adding 2 ladles full of broth from the pot (this helps temper the eggs so they don't curdle). Once fully combined, add to the pot and stir well. Remove from the heat immediately. Season with salt and pepper to taste. Stir in the parsley.
Mix the sparkling water and lemon. Serve over ice if desired.
Enjoy Greek lemon chicken rice soup with pita bread, dolmas, and lemon sparkling water.