Heat the oil in the Dutch oven over medium-high heat for 2 minutes. Sear the thighs skin-side down for 5 minutes and flip for 2 minutes, in two batches, until the skin is deep golden and lifts easily. Move the chicken to a plate.

Pour off all but 1 tablespoon of fat. Stir the onion, carrots, and celery for 7 to 8 minutes, adding the garlic for the last 30 seconds, until the onion is translucent and the garlic smells sweet.

Return the thighs and their juices to the pot. Add the water, broth, bay leaves, thyme, and remaining 1 teaspoon of salt, bring to a boil, and lower the heat until lazy bubbles break every second or two. Lift off the early foam.

Simmer with the lid cracked 1 inch for 35 to 40 minutes, until the thighs read at least 165°F (175°F is more tender), the meat pulls from the bone, and a carrot slides off a fork.

Cool the thighs on a cutting board for 5 minutes. Discard the skin and bones and shred the meat until it looks moist and glossy with no pink near the bone.

Skim the fat from the broth and remove the bay leaves and thyme stems, until the broth looks golden with only a few specks of fat.

Stir in the shredded chicken and lemon juice and simmer for 3 minutes. Add the parsley, until the soup steams gently and smells of thyme and lemon.