In a large pot, melt 2 tablespoons of butter over medium heat. Add finely chopped onion, sliced carrots, diced celery, and minced garlic. Stir the mixture occasionally and sauté for about 5 minutes until the vegetables are just softened and fragrant, turning a light golden color.

Pour in 4 cups of chicken broth and increase the heat to bring the mixture to a rolling boil. Once boiling, reduce the heat to low and let it simmer gently for 10 minutes.

Stir in 4 cups of broccoli florets into the pot. Continue to simmer the soup for another 10 to 15 minutes until the broccoli is tender and vibrant green.

For a creamier consistency, mix 2 tablespoons of cornstarch with 1 cup of half-and-half in a small bowl until smooth. Stir this mixture into the soup and allow it to cook for an additional 5 minutes.

Remove the pot from heat and immediately stir in 2 cups of shredded sharp cheddar cheese. Mix well until the cheese completely melts into the soup, creating a luscious and creamy finish.

Taste the soup and season with salt, freshly cracked pepper, and a pinch of paprika to enhance the flavor. Ladle the soup into bowls.