In a large pot, melt 2 tablespoons of butter over medium heat. Add 1 chopped onion, 1 diced carrot, and 2 minced garlic cloves, cooking for about 5 minutes until the onion is translucent and fragrant.

Stir in 3 diced potatoes and 4 cups of broth. Increase heat to high and bring to a boil, then reduce heat and let simmer for approximately 15 minutes or until fork-tender.

Add 4 cups of broccoli florets and cook for an additional 5 to 7 minutes until bright green and tender.

In a separate saucepan, melt 2 tablespoons of butter for the roux. Whisk in 2 tablespoons of flour and cook for about 1 minute. Gradually add 2 cups of milk, whisking until thickened.

Blend the soup in the pot using an immersion blender to your desired consistency.

Return the blended soup to the pot and stir in the roux, then gradually add in 2 cups of shredded cheddar cheese.

Season with salt, pepper, and a dash of paprika or cayenne to taste, then serve hot.