Melt Butter: In a medium saucepan over medium heat, melt butter.
Make Roux: Whisk in flour and cook for about 1 minute, stirring constantly, until you have a light golden roux.
Add Liquids: Gradually whisk in milk and beer, ensuring that there are no lumps. Keep stirring until mixture begins to thicken and comes to a gentle simmer.
Add Seasonings: Stir in Dijon mustard, garlic powder, onion powder, paprika, cayenne pepper, Worcestershire sauce, and a pinch of salt and pepper. Let mixture simmer for a couple of minutes.
Add Cheese: Reduce stovetop heat to low. Gradually add shredded cheeses, stirring constantly until the cheeses are completely melted and the sauce is smooth.
Adjust Consistency and Seasoning: If the dip is too thick, add a little more milk or beer. Taste and adjust the seasoning if necessary.
Serve: Remove saucepan from heat. Transfer dip to a serving bowl and sprinkle with fresh chives for garnish. (If the dip starts to thicken as it cools, reheat it gently while stirring, and add a splash of milk if necessary. Alternatively, transfer to a small slow cooker and keep on low temp to ensure the dip stays smooth and creamy.)