Start the Roux:
In a medium saucepan, melt the butter over medium heat. Sprinkle in the flour, whisking constantly, and cook for about 1 minute to eliminate any raw flour taste.
Add Beer and Milk:
Slowly pour in the beer and milk while whisking, smoothing out any lumps. Continue stirring and cook until the mixture starts to thicken, about 3-5 minutes.
Incorporate the Cheese:
Reduce heat to medium-low. Gradually add the cheddar and mozzarella cheeses, stirring until they’re completely melted and the mixture is smooth.
Season and Serve:
Stir in the Dijon mustard, garlic powder, and onion powder. Taste, then add salt and pepper as needed. Pour the dip into a serving bowl, garnish with chives or green onions if desired, and serve warm.