01 - Melt butter in a large pot over medium heat. Add diced onion and carrot, sauté until softened, approximately 5 minutes. Add minced garlic and cook for 1 additional minute until fragrant.
02 - Stir flour into the vegetable mixture, cooking for 1-2 minutes while stirring constantly to form a smooth roux and eliminate raw flour taste.
03 - Gradually whisk in vegetable broth, followed by whole milk and heavy cream, ensuring smooth incorporation without lumps.
04 - Add chopped broccoli stems and florets along with Dijon mustard and smoked paprika. Bring to a gentle simmer and cook uncovered for 15-20 minutes until broccoli becomes very tender.
05 - Use an immersion blender to puree soup to desired consistency, either completely smooth or slightly chunky. Alternatively, blend in batches using a countertop blender and return to pot.
06 - Reduce heat to low and gradually stir in grated sharp cheddar cheese until fully melted and incorporated. Season generously with salt and fresh black pepper to taste.
07 - Ladle hot soup into bowls and serve immediately, garnished with additional cheddar cheese or black pepper if desired.