Place water and chicken thighs in a pot. Bring to a boil; reduce heat and simmer until chicken is tender, about 30 minutes. Occasionally skim off scum as it forms.
Heat olive oil in a large pot over medium heat. Cook and stir carrots and onions until they begin to soften, 3 or 4 minutes. Stir in garlic and salt; cook for 1 minute.
Transfer chicken and cooking water to large pot. Add chicken broth. Bring mixture to a boil; reduce heat to low. Cover, leaving lid slightly ajar, and simmer at least 1 hour or up to 4 hours. Remove from heat.
Transfer chicken to a work surface. Remove skin, bones, and cartilage. Shred chicken meat using 2 forks. Return chicken to pot. Add broccoli, cauliflower, and zucchini. Bring back to a simmer over medium-high heat. Reduce heat so soup gently simmers. Cook until vegetables are tender, 20 to 30 minutes.