Prep the vegetables. Dice the onion, slice the carrots and celery, and mince the garlic.
Spiralize the zucchini into zoodles and set aside on a towel to absorb extra moisture.
Sauté the aromatics. Warm 1–2 tablespoons of olive oil in a large pot over medium heat. Add onion, carrots, and celery with a pinch of salt. Cook 6–8 minutes, stirring, until softened but not browned.
Add garlic and cook 30 seconds until fragrant.
Build the broth. Pour in the chicken broth. Add the bay leaf, a few sprigs of fresh thyme (or 1 teaspoon dried), and the Parmesan rind if using. Bring to a gentle simmer.
If using raw chicken, poach it now. Nestle raw chicken breasts or thighs into the simmering broth.
Cover and cook gently 12–18 minutes, until just cooked through. Remove, shred with two forks, and return to the pot.
If using cooked chicken, add it now. Stir in shredded rotisserie or leftover chicken and warm through for 3–4 minutes.
Season and brighten. Add lemon zest, a squeeze of lemon juice, and red pepper flakes if you like heat. Taste and adjust salt and pepper.
Remove the bay leaf and thyme stems.
Add the zoodles last. Drop the zucchini noodles into the hot soup and cook 1–3 minutes, just until tender. They should stay slightly crisp, not limp.
Finish with herbs. Stir in chopped parsley (and/or dill). Ladle into bowls and serve immediately.
Add more lemon at the table if you like.