Cook the vegetables. In a large Dutch oven, heat the olive oil over medium. When the oil shimmers, add the onion, celery, carrots, and garlic. Season with a good pinch of salt and pepper. Cook, stirring occasionally, until the veggies have softened, about 5 minutes.
Poach the chicken. Add the chicken, chicken stock, and water. Season with the oregano, thyme, bay leaf, red pepper flakes, and another dash of salt and pepper. Bring to a boil, then turn the heat to medium-low. Cook until the chicken is nearly fully cooked, 10 to 15 minutes, spooning out any foam that bubbles to the top of the cooking liquid.
Cook the rice. Transfer the chicken to a rimmed plate for now. Add the rinsed rice to the pot and raise the heat to bring the liquid back to a boil, then reduce the heat to medium-low and simmer until the rice is tender, 12 to 15 minutes or so.
Shred the chicken. While the rice is cooking, shred the chicken breasts into smaller pieces using two forks (hold one fork in each hand, tines facing down. Use one fork to pin the chicken in place, then drag the other fork through the meat, pulling away from the first.)
Finish and serve. Add the shredded chicken to the pot and simmer briefly, just to warm the chicken through. Remove and discard the bay leaf. Turn the heat off and stir in the lemon juice and zest, green onion, and fresh parsley. Taste and adjust the seasoning. Ladle into bowls and serve with lemon slices.
Notes
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How to Store Chicken and Rice Soup: Store leftover soup in an airtight container in the fridge for up to 4 days. The rice will keep absorbing broth as it sits, so the soup will thicken.
To Reheat: Warm gently on the stove over medium-low heat. Rice will naturally absorb the broth as it sits. Simply splash in a little extra chicken stock or water when reheating leftovers to bring it back to a soupy consistency.
To Freeze: The soup can be frozen for up to 3 months. For the best texture, freeze the soup without the lemon wheels and herbs and add them after reheating.