Prep the ingredients per the instructions above.
Mix the ingredients for the cucumber water in a pitcher. Refrigerate until ready to serve.
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, and then stir in the garlic.
Add the chicken broth (reserve some on the side in case it's needed) 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.
Remove the chicken from the pot and shred it apart very fine, using two forks to pull it apart. Then return the chicken to the pot.
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 and add more broth if needed to balance out the consistency.
Toast the bread then spread it with butter.
Portion the soup and serve with bread and cucumber water.