In a large pot, melt the butter over medium heat. Add the diced onion and sauté until translucent (about 5 minutes).
Add the minced garlic and cook for an additional minute until fragrant.
Stir in the broccoli florets and cover with the vegetable broth. Bring to a boil, then reduce the heat and simmer until the broccoli is tender (about 10 minutes).
Once the broccoli is cooked, remove the pot from heat. Using an immersion blender, blend the soup until smooth and creamy. Alternatively, you can blend in batches using a regular blender—just be careful with the hot liquid!
Return the blended soup to medium heat. Stir in the heavy cream, shredded cheddar cheese, and Dijon mustard. Mix until the cheese is melted and the soup is rich and creamy.