Melt 4 tablespoons butter in a large pot over medium heat
Add 1 diced onion and cook until softened
Add 2 minced garlic cloves and cook for 30 seconds
Stir in 1/4 cup all-purpose flour and cook for 1 to 2 minutes
Slowly whisk in 2 cups chicken broth and 2 cups milk
Add 4 cups chopped broccoli and 1 grated carrot
Simmer until the broccoli is tender, about 15 to 20 minutes
Reduce heat to low
Stir in 2 cups shredded cheddar cheese until melted
Season with salt, black pepper, and a pinch of paprika if desired
Serve hot