01 - In a large pot, melt the butter over medium heat. Add the onion and carrot, sautéing until softened, about 5 minutes. Add the garlic, and cook for 1 minute more.
02 - Stir in the flour and cook for 1-2 minutes, stirring constantly, to form a roux and remove raw flour taste.
03 - Gradually whisk in the vegetable broth, followed by the milk and cream, ensuring there are no lumps. Bring the mixture to a gentle simmer while whisking.
04 - Add the chopped broccoli stems and florets, Dijon mustard, and smoked paprika. Return to a gentle simmer. Cook uncovered for 15-20 minutes, until the broccoli is very tender.
05 - Use an immersion blender to puree the soup to your desired consistency (smooth or slightly chunky). Alternatively, blend in batches using a countertop blender, then return to the pot.
06 - Over low heat, stir in the 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.