01 - Melt butter in a large pot over medium heat. Add diced onion and carrot, sautéing until softened, approximately 5 minutes. Add minced garlic and cook for 1 additional minute until fragrant.
02 - Sprinkle flour into the pot, stirring constantly for 1-2 minutes to cook out the raw flour taste and form a smooth roux base.
03 - Gradually whisk in vegetable broth, followed by whole milk and heavy cream. Ensure continuous whisking to prevent lumps and achieve a smooth consistency.
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 completely tender.
05 - Use an immersion blender to puree soup to desired consistency, whether completely smooth or slightly textured. Alternatively, blend in batches using a countertop blender, then return to pot.
06 - Reduce heat to low and gradually stir in grated sharp cheddar cheese until fully melted and incorporated. Season generously with salt and freshly ground black pepper to taste.
07 - Ladle hot soup into bowls and serve immediately. Garnish with additional shredded cheddar or a sprinkle of black pepper if desired.