01 - Melt the butter in a large soup pot over medium heat. Add the diced onion and carrot, sautéing until softened and translucent, approximately 5 minutes. Stir in the minced garlic and cook for an additional minute until fragrant.
02 - Sprinkle the flour into the pot, stirring constantly to combine with the vegetables and butter. Cook for 1 to 2 minutes to eliminate the raw flour taste, creating a smooth roux base.
03 - Gradually whisk in the vegetable broth, followed by the whole milk and heavy cream. Ensure continuous whisking to prevent lumps and achieve a smooth, homogeneous mixture.
04 - Add the chopped broccoli stems and florets along with the Dijon mustard and smoked paprika. Bring the mixture to a gentle simmer and cook uncovered for 15 to 20 minutes, or until the broccoli is completely tender.
05 - Use an immersion blender to puree the soup directly in the pot until smooth, or leave slightly chunky if preferred. Alternatively, transfer to a countertop blender in batches, then return the pureed soup to the pot.
06 - Reduce the heat to low setting. Gradually stir in the grated sharp cheddar cheese until completely melted and fully incorporated into the soup. Avoid boiling to prevent separation.
07 - Taste the soup and season with salt and freshly ground black pepper as needed. Serve hot in bowls, optionally garnished with additional shredded cheddar or a sprinkle of black pepper.