In a large pot, melt the butter over medium heat. Add the chopped onion, celery, and carrot. Sauté for about 5 minutes, or until the vegetables are softened.
Stir in the minced garlic, paprika, dry mustard, and cayenne pepper. Cook for an additional 1-2 minutes until fragrant.
Sprinkle the all-purpose flour over the vegetables and stir to create a roux. Cook for 2-3 minutes, or until the roux begins to turn golden brown.
Gradually whisk in the chicken or vegetable broth, beer, and heavy cream. Continue to whisk until the mixture is smooth.
Reduce the heat to low and add the shredded cheddar and mozzarella cheese, stirring until the cheese is fully melted and the soup is smooth and creamy.
Season the soup with salt and black pepper to taste. Simmer for 5-10 minutes, stirring occasionally, until it thickens to your desired consistency.