Heat oil over medium heat in a large pot. Once shimmering, add the onion, celery and carrots and saute until soft, about 8 minutes. Add garlic and saute until fragrant, about 1-2 minutes.
Add the broth, chicken, salt, pepper, and rosemary. Bring broth to a boil, then reduce to a simmer and continue to cook until the chicken is cooked through (internal temp should reach 165 F), about 22-26 minutes. Make sure the chicken is fully submerged as you simmer.
Remove soup from heat. Use tongs to remove the cooked chicken to a cutting board, and once it's cool enough to handle, dice it into bite-sized pieces.
Add the lemon juice, lemon zest, zucchini noodles, chicken breast, and parmesan if using to the soup, and stir to combine. Ladle into bowls, and enjoy!