Start by chopping the cauliflower and broccoli into small florets. Set these aside for later.
Heat the butter in a large pot over medium heat. Add the diced onion and sauté until translucent, about 5 minutes. Then, add the minced garlic and sauté for another 1-2 minutes until fragrant.
Add the cauliflower and broccoli florets to the pot. Pour in the vegetable or chicken broth and bring to a boil. Reduce the heat to a simmer, cover the pot, and cook until the vegetables are tender, about 15 - 20 minutes.
Using an immersion blender, blend the soup until smooth. If you prefer a chunkier texture, you can blend only part of the soup. Alternatively, you can carefully transfer the soup in batches to a blender and blend until smooth, then return it to the pot.
Stir in the heavy cream or milk. Gradually add the shredded cheddar cheese, stirring continuously until melted and well combined. Season with salt, and pepper.