Freshly grate the cheeses from blocks. Don't use shredded cheese from the bag. It does not melt as smoothly.
Melt butter in a medium saucepan over medium heat. Add flour, mustard, garlic powder, salt, and cayenne and whisk for 30 seconds.
Slowly pour in the beer and milk, whisking constantly. Simmer for 5-7 minutes, whisking often, until thickened.
Reduce the heat to low. Add the grated cheese a handful at a time, whisking between each addition, until the cheese is fully melted. Serve immediately.
Notes
Store beer cheese dip in the refrigerator for up to 3 days.Reheat beer cheese dip on the stove over low heat. Stir frequently until heated through.Mix-ins: 1/4 cup chopped onion and/or 1/4 cup chopped jalapeño may be sautéed in the melted butter for a more spicy flavor.