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 eliminate raw flour taste.
03 - Gradually whisk in vegetable broth followed by milk and cream, ensuring no lumps form. Continue whisking until fully incorporated and smooth.
04 - Add chopped broccoli stems and florets along with Dijon mustard and smoked paprika. Bring to a gentle simmer and cook uncovered for 15-20 minutes until broccoli is very tender.
05 - Use an immersion blender to puree soup to desired consistency, whether completely smooth or slightly chunky. Alternatively, blend in batches using a countertop blender, then return to pot.
06 - Over low heat, stir in grated cheddar cheese until melted and fully incorporated. Season generously with salt and pepper to taste.
07 - Serve hot in bowls, garnished with extra shredded cheddar or a sprinkle of black pepper if desired.