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In a large, heavy-bottomed pot over medium heat, melt butter and add onions and pepper. Sauté vegetables until tender, about 6 minutes. Stir in garlic and mustard powder and cook 1 minute more. Sift in flour and stir with a wooden spoon for about 3 minutes, coating vegetables evenly. Switch to a whisk and gradually pour in milk while whisking. Add chicken stock and beer and bring up to a simmer, stirring often. Once simmering, stir in cheddar cheese and reduce heat to medium-low, stirring until melted. Add cayenne, if using and season to taste. Ladle into bowls and garnish with cheese.

Soup can be prepared ahead and chilled, but take care to reheat over low heat, stirring often.