Begin by melting the butter in a medium saucepan over medium-low heat. Toss in the diced onion and sauté until it becomes soft and translucent, which should take around 4 to 6 minutes. Slowly incorporate the flour into the mixture, ensuring you whisk constantly until it takes on a light brown hue, usually about 1 minute.
Gradually pour in the beer and half and half while whisking continuously. Let the mixture cook until it thickens and starts to bubble gently, which should take approximately 5 to 7 minutes. Once thickened, reduce the heat to low and slowly add in the cheeses while continuing to whisk. Keep whisking until the cheeses are fully melted and the mixture is well combined. Remove the saucepan from the heat.
Stir in the Dijon mustard and Worcestershire sauce until fully incorporated. For added flavor, garnish with chopped chives and a dash of black pepper if desired. Serve the cheese sauce alongside pretzels, reheating gently as needed.
