In a small sauce pot set over medium-low heat, cook the beer and milk just until warmed through but not boiling.
In a separate medium sauce pot set over medium-low heat, melt the butter. Whisk in the flour and cook, whisking, until the mixture is slightly toasted, about 2 minutes. Whisk in the Dijon mustard and smoked paprika.
Slowly stream in the warm beer mixture, whisking constantly. Stir in the Worcestershire and grated cheese and and cook, whisking frequently, until the cheese is completely melted and the mixture is smooth. Serve immediately.