Melt butter in large pot over medium heat.
Add carrot, celery, onion, and garlic. Sauté for 10 minutes or until softened.
Add flour and stir well. Cook, stirring frequently, for 3 minutes, or until flour turns golden brown.
Slowly pour milk into pot, whisking until combined.
Reduce heat to medium-high. Stir in beer and mustard. Bring to a boil, whisking constantly until foam settles.
Simmer on low for 10 minutes, or until thickened. Remove from heat and whisk in cheese in portions.
Add salt and pepper to taste. Serve immediately.