Melt the butter in a large pot over medium heat until it bubbles gently, about 1 minute.

Sauté the finely chopped onion for 3 minutes until translucent, then add minced garlic and cook for 30 seconds.

Sprinkle flour over the onion and garlic, stirring constantly for 1 minute until the mixture resembles a paste.

Whisk in the chicken broth and milk gradually, bringing to a gentle simmer and stirring frequently for about 5 minutes.

Add broccoli florets (and shredded carrots if using) and simmer for another 5 to 7 minutes until broccoli is tender.

Reduce heat, add shredded cheddar cheese, stirring continuously until melted, then stir in heavy cream.

Serve into bowls, garnishing with extra cheese and black pepper if desired.