01 - Melt butter in a large pot over medium heat. Add onion and carrot, sautéing until softened, about 5 minutes. Add garlic and cook for 1 minute more.
02 - Stir in flour and cook for 1-2 minutes, stirring constantly, to form a roux.
03 - Gradually whisk in vegetable broth, followed by milk and cream, ensuring there are no lumps.
04 - Add 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 the soup to your desired consistency. Alternatively, blend in batches using a countertop blender, then return to the pot.
06 - Over low heat, stir in grated cheddar cheese until melted and fully incorporated. Season with salt and pepper to taste.
07 - Serve hot, garnished with extra cheddar or a sprinkle of black pepper if desired.