01 - In a large soup pot, melt the butter over medium heat. Add the diced onion and carrot, sautéing until softened and fragrant, approximately 5 minutes. Add the minced garlic and continue cooking for 1 additional minute.
02 - Sprinkle the flour into the pot while stirring constantly. Cook the flour for 1 to 2 minutes to form a smooth roux, eliminating any raw flour taste.
03 - Gradually whisk in the vegetable broth, followed by the whole milk and heavy cream. Whisk continuously to ensure the roux incorporates smoothly without forming lumps.
04 - Add the chopped broccoli stems and florets along with the Dijon mustard and smoked paprika. Bring the mixture to a gentle simmer. Cook uncovered for 15 to 20 minutes until the broccoli becomes completely tender.
05 - Using an immersion blender, puree the soup directly in the pot until reaching your preferred consistency—completely smooth or with some texture remaining. Alternatively, transfer to a countertop blender and process in batches before returning to the pot.
06 - Reduce the heat to low. Gradually stir in the grated sharp cheddar cheese, allowing each addition to melt completely before adding more. Continue stirring until the cheese is fully incorporated and the soup is velvety smooth. Season generously with salt and freshly ground black pepper.
07 - Ladle the hot soup into serving bowls. Garnish with additional shredded cheddar cheese or a light sprinkling of black pepper if desired. Serve immediately while hot.