Heat the olive oil in a large pot over medium heat. Add the chopped onion and sauté until translucent, about 3-4 minutes.

Stir in the minced garlic and cook for an additional minute until fragrant.

Add the diced potatoes and vegetable broth to the pot. Bring to a boil, then reduce heat and let simmer for about 10 minutes until the potatoes are tender.

Add the broccoli florets and cook for another 5-7 minutes until tender but still bright green.

Using an immersion blender, blend the soup to your desired consistency. For a chunkier texture, blend only half of the soup.

Return the pot to low heat and stir in the milk and shredded cheddar cheese. Stir until the cheese is melted and the soup is creamy.

Season with salt, pepper, and dried thyme if using. Taste and adjust seasoning as needed.

Ladle the soup into bowls and garnish with chopped fresh parsley. Serve hot.