Melt the butter in a stockpot or Dutch oven over medium heat.
Add the chopped onion and sauté until softened and translucent, about 3-4 minutes.
Stir in the minced garlic and cook for another minute until fragrant.
Pour in the vegetable broth and add the frozen broccoli florets. Increase the heat to high and bring the soup to a gentle boil.
Once boiling, reduce the heat to maintain a steady simmer.
Cook for about 15 minutes, or until the broccoli is very tender.
Reduce the heat to the lowest possible setting. This is crucial to prevent the soup from curdling. Slowly stir in the heavy cream, followed by the shredded cheddar cheese.
Continue stirring gently until the cheese is completely melted and the soup is smooth and creamy.
Taste the soup and season generously with salt and pepper. Ladle into bowls and serve immediately, garnished with a sprinkle of extra cheddar cheese.
Shred your own cheese! Pre-shredded cheeses are often coated with anti-caking agents that can make the soup grainy. A block of sharp cheddar will melt much more smoothly.