Melt Butter: In a large pot, melt butter over medium heat.
Cook Vegetables: Add finely chopped onion, carrots, and celery. Cook for about 10 minutes, stirring occasionally, until vegetables are softened.
Add Garlic: Stir in minced garlic and cook for an additional 1 minute until fragrant.
Make Roux: Sprinkle all-purpose flour over vegetables and stir well to combine. Cook for about 2 minutes, stirring constantly, until flour is lightly browned.
Add Liquids, Boil, Then Simmer: Gradually pour in chicken broth then beer, stirring constantly to avoid lumps. Bring mixture to a boil, then reduce stovetop heat to reach a gentle simmer, usually medium-low or low. Cook for about 15 minutes, stirring occasionally, until soup thickens.
Add Half-and-Half: Stir in half-and-half and cook for another 5 minutes.
Melt Cheese: Gradually add shredded cheese, stirring constantly until cheese is completely melted and soup is smooth.
Season Soup: Stir in Worcestershire sauce and Dijon mustard. Season with salt and pepper to taste. Cook for an additional 5 minutes to let flavors meld.
Serve Soup: Ladle soup into bowls and garnish with chopped chives or green onions and croutons if desired.