Melt the butter in a small saucepan over low heat. Add the flour to create the roux (a paste), whisking constantly for 1 minute until the paste is thickened and bubbly.
Slowly whisk in the milk and beer, stirring until completely smooth and the mixture has thickened, about 1 minute. Stir in remaining ingredients except for the cheese.
Slowly add the cheese and whisk until completely melted and smooth. Remove from heat and serve immediately while warm.