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 until fragrant.
02 - Stir flour into the pot 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 milk and cream, ensuring no lumps form. Continue whisking until fully incorporated.
04 - Add chopped broccoli stems, florets, Dijon mustard, and smoked paprika. Bring to a gentle simmer and cook uncovered for 15-20 minutes until broccoli is very tender.
05 - Use immersion blender to puree soup to preferred texture (smooth or slightly chunky). Alternatively, blend in batches using countertop blender, then return to pot.
06 - Reduce heat to low and 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 sprinkle of black pepper if desired.