In a large pot, melt 2 tablespoons of butter over medium heat. Add 1 cup of chopped onion and sauté for about 3 minutes until soft and translucent.

Stir in 2 minced garlic cloves and continue cooking for an additional 30 seconds.

Sprinkle 2 tablespoons of all-purpose flour over the onion-garlic mixture, stirring until it forms a paste. Cook for about 1 minute.

Gradually whisk in 2 cups of chicken broth and 1 cup of milk. Increase heat to bring to a gentle simmer, allowing it to cook for around 5 minutes.

Introduce 4 cups of broccoli florets and optional 1 cup of shredded carrots into the pot. Continue to simmer for another 5 to 7 minutes.

Reduce heat to low and gradually stir in 2 cups of shredded sharp cheddar cheese, and optional 1/2 cup of heavy cream.

Ladle the soup into bowls and garnish with extra cheese, black pepper, and fresh parsley if desired.