Prep the ingredients per the instructions above.
Mix the ingredients for the cucumber pomegranate 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, 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.
Heat the broiler on low.
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.
Make the pita: heat the oil in a nonstick oven-safe skillet. Warm the pita on both sides. Then top with cheese and place under the broiler to melt. Once melted, cut into bite-sized pieces.
Portion the soup and serve with pita and cucumber pomegranate water.