01 - Melt the butter in a large pot over medium heat. Add the diced onion and carrot, sautéing until softened, about 5 minutes. Add the minced garlic and cook for 1 minute more until fragrant.
02 - Stir the flour into the vegetables and cook for 1-2 minutes, stirring constantly to form a smooth roux and eliminate raw flour taste.
03 - Gradually whisk in the vegetable broth, followed by the milk and cream, ensuring no lumps form. Continue whisking until fully incorporated.
04 - Add the chopped broccoli stems and florets, Dijon mustard, and smoked paprika. Bring to a gentle simmer. Cook uncovered for 15-20 minutes until broccoli is very tender.
05 - Use an immersion blender to puree 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. Stir in the grated cheddar cheese until melted and fully incorporated. Season with salt and pepper to taste.
07 - Ladle hot soup into bowls. Garnish with extra cheddar cheese or a sprinkle of black pepper if desired.