In a large pot over medium heat, melt 3 tablespoons of unsalted butter until bubbling. Add diced onion, chopped carrot, chopped celery, and minced garlic, and sauté for about 5 minutes until the onion is translucent and the vegetables are tender.

Sprinkle 1/4 cup of all-purpose flour over the sautéed vegetables, stirring constantly for about 2 minutes to prevent a raw taste.

Gradually whisk in 4 cups of low-sodium broth, ensuring no lumps remain. Once combined, add chopped broccoli, smoked paprika, and season with black pepper and kosher salt to taste. Bring to a simmer.

Cover the pot and let the soup simmer for 15 minutes, stirring occasionally.

Reduce heat to low and stir in 1 cup of whole milk, 1/2 cup of heavy cream, and 8 ounces of cream cheese. Mix until smooth.

Gradually add in 2 cups of shredded sharp cheddar cheese, stirring until melted.

For a smoother texture, blend with an immersion blender, leaving some chunks.

Taste and adjust seasoning as needed, and serve hot with optional toppings.