01 - Melt the unsalted butter in a large soup pot over medium heat. Add the diced onion, carrot, celery, and minced garlic. Sauté for about 5 minutes, stirring occasionally, until vegetables are tender.
02 - Sprinkle the all-purpose flour evenly over the sautéed vegetables. Cook, stirring constantly, for 2 minutes to eliminate any raw flour taste.
03 - Gradually whisk in the vegetable or chicken broth to avoid lumps. Add the chopped broccoli, smoked paprika, salt, and black pepper. Bring the mixture to a simmer.
04 - Cover the pot and cook for 15 minutes, or until the broccoli is fork-tender.
05 - Reduce heat to low. Stir in the whole milk, heavy cream, and softened cream cheese until fully incorporated and smooth. Simmer gently for 5 minutes.
06 - Gradually add shredded sharp cheddar cheese, stirring until completely melted and the soup is creamy.
07 - For a smoother consistency, use an immersion blender to blend the soup partially, leaving some texture if desired.
08 - Taste the soup and adjust seasoning with additional salt and pepper if needed. Ladle hot soup into bowls. Garnish with extra cheddar cheese, crumbled bacon, chopped green onions, and a dollop of sour cream as preferred.