In a large pot, heat the olive oil over medium heat. Add the diced onion and sauté until translucent, about 5 minutes.
Add the minced garlic and thyme to the pot, cooking for an additional 1-2 minutes until fragrant.
Stir in the chopped broccoli and vegetable broth. Bring to a boil, then reduce heat and simmer for about 15-20 minutes, or until the broccoli is tender.
Use an immersion blender to puree the soup until smooth; alternatively, you can transfer the soup in batches to a blender.
Once blended, return the soup to the pot and stir in the heavy cream and shredded cheddar cheese until the cheese is melted and fully incorporated.
Season with salt and pepper to taste. If the soup is too thick, you can thin it out with a little more vegetable broth or water.
Let the soup simmer for an additional 5 minutes to warm through.
Notes
Serve the soup in bowls, garnished with croutons and a sprinkle of additional cheddar cheese if desired.