In a large saucepan over medium heat, stir together carrots, onion, celery, and garlic. Add cayenne pepper, salt/pepper, chicken broth and beer; simmer until vegetables are tender, about 12 minutes. Remove from heat.
Heat butter in another large soup pot over medium-high heat. Whisk in flour and cook stirring until the flour is light brown, about 3 or 4 minutes. Whisk in milk until thickened and remove from heat, then gradually stir in cheese and keep warm. Add mashed potatoes if you’d like, that would be even better in my eyes.
Stir the beer mix into the cheese mix and add the Dijon mustard, Worcestershire sauce, and dry mustard. Add optional hot pepper sauce. Bring to a simmer, and cook 10 minutes.