01 - Melt butter in a large soup pot over medium heat. Add diced onion and carrot, sautéing until softened and fragrant, approximately 5 minutes. Stir in minced garlic and cook for 1 additional minute until aromatic.
02 - Sprinkle flour into the pot while stirring constantly. Cook the roux for 1-2 minutes to eliminate raw flour taste, stirring continuously to prevent scorching.
03 - Gradually whisk in vegetable broth, followed by whole milk and heavy cream. Ensure smooth incorporation by whisking vigorously to eliminate any lumps from the roux.
04 - Add chopped broccoli stems and florets along with Dijon mustard and smoked paprika. Bring mixture to a gentle simmer and cook uncovered for 15-20 minutes until broccoli becomes completely tender.
05 - Use an immersion blender to puree soup to desired consistency, either completely smooth or retaining slight texture. Alternatively, blend in batches using a countertop blender, then return to pot.
06 - Reduce heat to low setting. Gradually stir in grated sharp cheddar cheese until fully melted and incorporated throughout the soup. 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 cheese or a fresh grind of black pepper if desired.