Melt butter in a medium saucepan over medium-low heat. When the butter has melted and begins to foam, add the flour and whisk; cook 2 to 3 minutes to cook the rawness out of the flour. Add garlic, horseradish, paprika. Whisk to combine.
Slowly add beer a little at a time, whisking during each incorporation. The beer will foam up a little and it may seem like it will not stop, but it will. Whisk until smooth.
Add in half & half, Worcestershire and cheese; whisk it has melted and the mixture is smooth. Keep warm until ready to serve, whisking occasionally.
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