Melt the butter in a medium saucepan over medium heat until it begins to foam.
Whisk in the flour and cook for 1-2 minutes, stirring constantly, until the mixture is golden and smells slightly nutty.
Gradually pour in the beer and milk while whisking continuously to prevent lumps from forming, then bring the mixture to a gentle simmer.
Reduce heat to low and add the shredded cheddar and smoked gouda in three batches, stirring until each addition is completely melted before adding more.
Stir in the Dijon mustard, smoked paprika, garlic powder, and cayenne pepper until well combined.
Season with salt and black pepper to taste, then transfer to a serving bowl or keep warm in a slow cooker set to low.
Serve immediately with soft pretzels, tortilla chips, crackers, or fresh vegetables for dipping.