Pat the chicken leg quarters dry and season both sides generously with salt and pepper. Heat the olive oil in a large pot or Dutch oven over medium heat. When the oil is shimmering, add the chicken, skin-side down, in an even layer. Cook, undisturbed, until deeply golden brown, 4 to 5 minutes. Flip and repeat on the other side, then transfer the chicken to a plate.

Add the onions, carrots, and celery to the fat remaining in the pot and season with 1 teaspoon salt and ¼ teaspoon pepper. Cook over medium heat, stirring occasionally, until the vegetables begin to soften and smell fragrant, 8 to 10 minutes.

Return the chicken to the pot along with any accumulated juices. Add the water and the rice, seasoning again with more salt and pepper. Stir and bring the soup to a boil, then reduce the heat to medium-low and simmer, stirring occasionally, until the stock is flavorful and the rice is very tender, about 45 minutes. If the soup begins to look too thick, simply add a splash more water.

Turn off the heat. Remove the chicken from the soup and transfer it to a cutting board. When the chicken is cool enough to handle, remove and discard the skin and pull the meat from the bones. Chop or shred the meat into bite-sized pieces.

Return the shredded chicken to the pot, add the remaining 1 teaspoon salt, and stir to combine. Turn the heat up to medium-low to warm the soup through again, then taste and adjust the seasoning with additional salt and pepper as needed. Finish with fresh parsley and or lemon juice, if using.