Melt the butter in a large pot or Dutch oven over medium heat. Add the onion and cook for 3 to 5 minutes, until soft. Add the garlic and cook 1 minute, stirring. Add the broccoli, potatoes, carrot and broth.
Bring to a boil and then reduce heat to a low simmer. Cover and cook 10 to 15 minutes, until the vegetables are tender.
Remove from heat and stir in the cheese until melted. Using an immersion blender, puree the soup. We like to puree it only partway, so that some pieces of potato and broccoli remain. (Alternatively, you can transfer the soup in batches to a blender to puree.)
Season soup with salt and pepper to taste and serve.
Notes
If you are preparing this recipe as gluten-free, just be sure to use gluten-free broth.