In a large pot, melt the butter over medium heat.

Add the diced onion and minced garlic to the pot and sauté until the onion is translucent.

Add the broccoli florets and vegetable broth to the pot. Bring to a boil, then reduce heat and simmer for 15-20 minutes, or until the broccoli is tender.

Using an immersion blender, puree the soup until smooth. Alternatively, you can transfer the soup to a blender and blend until smooth, then return to the pot.

Stir in the milk and cheddar cheese until melted and creamy. Season with salt and pepper to taste.

Serve hot and enjoy!