Heat beer and next 5 ingredients in a large skillet over medium-low heat. Simmer for 10 minutes. Add fish fillets in a single layer and simmer for 10 minutes (or a minute or two less for fillets less than 1-inch thick). Transfer fish to a well-greased baking dish.

Melt butter in a skillet over medium heat and stir in flour, a little at a time, until blended with the butter. Continue stirring until it is smooth. Add milk, a little at a time, to the flour and butter mixture, while stirring. Add cheese and salt and pepper to taste. Reduce heat to medium-low and continue stirring until sauce is hot and smooth.

Pour sauce over fish in baking dish. Place baking dish under a pre-heated broiler and broil until sauce is lightly browned and bubbly.