Heat the butter in a large pot over medium heat.
Add the onions, celery, carrots, and garlic and let saute until softened, about 5 minutes. Stir frequently to prevent sticking.
Add the flour, stirring to coat the vegetables, and cook for 3 more minutes.
Next, add the broth and beer, stirring well. Simmer over medium heat for 10 minutes.
Slowly stir in the milk and mix well, let heat up for 1 minute.
Turn the heat off but leave the pot on the burner. Slowly add the cheese, stirring to incorporate between each handful.
Add the mustards, salt, and peppers here.
Once the cheese is melted, use an immersion blender to whip the soup into a velvety, smooth mixture, for about 2-3 minutes. Taste again and adjust seasonings.
Serve in a bread bowl, with bread or a big soft pretzel or garlic knot on the side if you wish. Or, put the garlic knot IN the bowl and cover with soup for a real treat.