Finely dice the onion, carrots, and celery. Mince the garlic. Chop the broccoli florets into small, bite-sized pieces. Grate the sharp cheddar cheese.

In a large pot or Dutch oven, melt the butter over medium heat. Add the diced onion, carrots, and celery. Sauté for 8-10 minutes, or until the vegetables are softened and the onion is translucent.

Add the minced garlic and cook for another minute until fragrant.

Sprinkle the all-purpose flour over the sautéed vegetables. Stir constantly for 1-2 minutes to cook out the raw flour taste, creating a roux.

Gradually whisk in the whole milk, a little at a time, ensuring no lumps form. Once all the milk is incorporated, slowly whisk in the chicken or vegetable broth. Bring the mixture to a gentle simmer, stirring occasionally.

Stir in the Dijon mustard, salt, black pepper, and ground nutmeg. Continue to simmer for about 5 minutes, allowing the soup to slightly thicken.

Add the broccoli florets to the simmering soup. Cook for 10-12 minutes, or until the broccoli is tender. If using frozen broccoli, it might take a couple of minutes longer.

Using an immersion blender, carefully blend about half to two-thirds of the soup directly in the pot, leaving some broccoli chunks for texture. Alternatively, carefully transfer a portion of the soup to a standard blender, blend until smooth, and then return it to the pot.

Remove the pot from the heat. This is crucial for preventing the cheese from clumping. Gradually stir in the freshly grated sharp cheddar cheese, a handful at a time, whisking continuously until it is completely melted and smooth. Do not bring the soup back to a boil once the cheese is added, as this can make the cheese grainy.

Taste the soup and adjust seasoning with more salt or pepper if needed. Ladle hot soup into bowls and garnish with extra shredded cheddar and fresh parsley, if desired. Serve immediately.