Use a large pot to melt the butter over medium heat. Once melted, add the garlic and sauté for a few minutes until it becomes aromatic.
Slowly pour in the heavy cream and allow it to come to a gentle simmer.
Next, introduce the vegetable or chicken broth to the pot and return to a simmer.
Combine the broccoli, diced potato, and grated carrot into the simmering mixture. Let them cook for about 25 to 30 minutes, or until the broccoli softens.
Add the shredded cheddar cheese to the pot, stirring continuously until the cheese has completely melted and blended into the soup.
Season the soup with salt and pepper according to your taste preferences.
Serve the soup warm, optionally topped with more shredded cheese for garnish.