01 - Melt butter in a large soup pot over medium heat. Add diced onion and carrot, sauté until softened and fragrant, approximately 5 minutes. Stir in minced garlic and cook for an additional minute until aromatic.
02 - Sprinkle flour evenly over the sautéed vegetables. Stir constantly for 1-2 minutes to cook out the raw flour taste and create a smooth roux base that will thicken the soup.
03 - Slowly whisk in vegetable broth, followed by whole milk and heavy cream. Whisk continuously to prevent lumps and ensure a smooth, homogeneous mixture.
04 - Add broccoli florets and chopped stems along with Dijon mustard and smoked paprika. Bring the mixture to a gentle simmer, then cook uncovered for 15-20 minutes until broccoli is completely tender.
05 - Use an immersion blender to puree directly in the pot until reaching your preferred consistency, either completely smooth or slightly chunky. For a countertop blender, work in batches and return pureed soup to the pot.
06 - Reduce heat to low setting. Gradually stir in grated sharp cheddar cheese, allowing each addition to melt completely before adding more. Season generously with salt and freshly ground black pepper.
07 - Ladle hot soup into bowls and serve immediately. Garnish with additional shredded cheddar or a grinding of fresh black pepper if desired.