Step 1: Prepare the Beer Cheese Sauce. In a medium saucepan over medium heat, melt the butter. Whisk in the flour and cook for 1-2 minutes, stirring constantly, to create a roux.
Step 2: Slowly whisk in the milk until smooth, then gradually add the beer, continuing to whisk until the mixture thickens slightly.
Step 3: Reduce heat to low. Add the shredded cheddar and Gruyere (if using) cheeses a handful at a time, whisking constantly until each addition is fully melted and smooth before adding more. Stir in garlic powder, dry mustard, paprika, salt, and pepper to taste. Keep warm over very low heat, stirring occasionally.
Step 4: Form the Burger Patties. Divide the ground beef into 4 equal portions. Gently form each portion into a 1/2-inch thick patty, making a slight indentation in the center of each to prevent bulging. Season generously with salt and black pepper on both sides.
Step 5: Cook the Burgers. Heat a grill or a large cast-iron skillet over medium-high heat. Cook patties for 4-5 minutes per side for medium-rare, or longer for desired doneness.
Step 6: Toast the Buns. While the burgers cook, slice the pretzel buns in half and lightly toast them on the grill or in a separate pan until golden brown.
Step 7: Assemble the Burgers. Place the bottom half of each toasted pretzel bun on a plate. Spoon a generous amount of warm beer cheese sauce onto the bun. Add a cooked burger patty, then spoon more beer cheese sauce over the patty. Top with desired optional toppings and the other half of the pretzel bun. Serve immediately.