In a large saucepan over medium heat, stir together carrots, onion, celery, and garlic. Stir in hot pepper sauce, cayenne pepper, salt, and pepper. Pour in chicken broth and beer; simmer until vegetables are tender, about 12 minutes. Remove from heat.
Meanwhile, heat butter in a large soup pot over medium-high heat. Stir in flour with a wire whisk; cook, stirring until the flour is light brown, about 3 or 4 minutes. Gradually stir in milk, whisking to prevent scorching, until thickened. Remove from heat, and gradually stir in cheese. Keep warm.
Stir beer mixture into cheese mixture. Stir in Dijon mustard, Worcestershire sauce, and dry mustard. Adjust for hot pepper sauce. Bring to a simmer, and cook 10 minutes.
