– Prepare the Vegetables: Start by chopping the broccoli into small florets. In a separate bowl, chop the onion, celery, and carrot. – Sauté the Vegetables: In a large pot, melt the butter over medium heat. Add the chopped onion, celery, and carrot. Sauté until the vegetables become tender. – Create the Roux: Sprinkle the all-purpose flour over the sautéed vegetables and stir continuously for a few minutes to create a roux, which will thicken the soup. – Add Broth and Broccoli: Gradually whisk in the chicken or vegetable broth, ensuring a smooth mixture. Add the chopped broccoli florets to the pot. – Simmer to Perfection: Allow the soup to simmer gently for about 15-20 minutes or until the broccoli is tender and cooked to your liking. – Blend Smooth: Using an immersion blender or a regular blender (in batches, if needed), puree the soup until it’s smooth and creamy. – Incorporate Milk and Cheese: Return the soup to the pot and pour in the whole milk. Stir in the shredded sharp cheddar cheese, allowing it to melt and create that luscious cheese flavor. – Season and Serve: Season your Broccoli Cheese Soup with salt and pepper to taste. Ladle it into bowls and, if desired, garnish with additional shredded cheese, crumbled bacon, or chopped chives.