Heat the olive oil in a large pot or Dutch oven over medium heat. Add the chopped onion and cook until translucent, about 5 minutes.
Stir in the minced garlic, diced carrots, and diced celery. Cook for an additional 5 minutes, until the vegetables begin to soften.
Add the diced zucchini and potato to the pot, followed by the vegetable broth. Bring to a boil, then reduce heat to a simmer. Cook for about 15 minutes, or until all the vegetables are tender.
Once the vegetables are cooked, use a whisk to stir in the whole milk, followed by the shredded cheddar cheese. Stir continuously until the cheese is melted and the soup is creamy.
Season the soup with dried thyme, salt, and pepper to taste. Adjust seasoning as necessary.
Serve hot, garnished with fresh parsley if desired.