Melt butter in a small saucepan over medium heat. Stir in flour and cook for 2 minutes. Whisk in beer and cook until the mixture begins to thicken and bubble, about 4 minutes. Whisk in cheese and cook, stirring, until cheese has completely melted. Add mustard and salt.

In a small bowl, whisk the egg until lemony yellow. Pour a small amount of the cheese mixture into the bowl, whisking vigorously, to temper the egg. Pour the egg mixture into the pot and cook for 2 minutes, whisking constantly, until thick.

Remove from heat and continue to whisk vigorously for a couple of minutes. If, despite your best efforts, the egg curdles, strain the dip through a sieve before serving. Serve warm with pumpernickel pretzels or crusty bread and a cold beer.