In a large stock pot over medium heat, melt the butter. Stir in the onion, salt, and pepper. Stir and cook for 8 minutes, until the onion is soft and translucent. Stir in the garlic and cook for another minute.

Sprinkle the flour over the onions and stir continuously for 3-4 minutes. You want the flour to thicken and turn a nice golden brown. Slowly pour in the milk. Using a whisk, mix the milk into the flour while you're pouring and then keep stirring for 2 minutes.

Stir in the broth, broccoli, carrot, and mustard. Mix well and bring to a boil. Lower the temp to a simmer. Cover and cook for 45 minutes, or until broccoli and carrots are tender.

Remove the soup from the heat and use an immersion blender to blend the soup. Blend it until you get to the consistency you want, whether super creamy or kind of chunky. Then put the soup pot back on low heat and stir in the cheese. Keep stirring until all the cheese in melted. Season with salt and pepper. Serve immediately with croutons.