In a large pot, heat the olive oil over medium heat. Add the chopped onion and sauté until translucent, about 5 minutes.
Add the minced garlic and cook for another minute until fragrant.
Stir in the diced potatoes and vegetable broth, then bring to a boil. Reduce heat and simmer for 10 minutes, or until the potatoes are tender.
Add the broccoli florets and thyme to the pot. Cook for an additional 5-7 minutes until the broccoli is tender but still bright green.
Using an immersion blender, blend the soup until smooth. Alternatively, you can transfer to a blender in batches (be careful with hot liquids!).
Return the blended soup to the pot and stir in the milk. Heat through on low heat, then add the shredded cheddar cheese in increments, stirring continuously until melted and creamy.
Season with salt and pepper to taste, and adjust thickness with water or broth if necessary.
Serve hot, garnished with fresh parsley if desired.