01 - Melt butter in a large pot over medium heat. Add diced onion and carrot, sautéing until softened, about 5 minutes. Add minced garlic and cook for 1 minute more until fragrant.
02 - Stir flour into the vegetable mixture and cook for 1-2 minutes, stirring constantly, to form a smooth roux and remove raw flour taste.
03 - Gradually whisk in vegetable broth, followed by whole milk and heavy cream, ensuring no lumps form. Bring the mixture to a gentle simmer while stirring continuously.
04 - Add chopped broccoli stems and florets, Dijon mustard, and smoked paprika to the pot. Simmer uncovered for 15-20 minutes until broccoli is very tender and easily pierced with a fork.
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, then return to pot.
06 - Reduce heat to low and gradually stir in grated sharp cheddar cheese until fully melted and incorporated. Season with salt and black pepper to taste.
07 - Ladle hot soup into bowls and serve immediately. Garnish with additional shredded cheddar or a sprinkle of black pepper if desired.