Start a wood campfire and wait until the flames die down and the coals glow red with white ash. Lower a grilling grate so that it sits above the coals and you have nice, even medium heat.
Place the butter and garlic in a large cast-iron skillet and set over the fire (or on a stovetop over medium heat). As the butter melts, the garlic will infuse the butter and become fragrant. Stir well and season with salt and pepper; transfer the garlic butter to a small bowl and set aside to cool.
Place the same skillet back over medium heat to warm up. Add enough canola oil to coat the skillet, about 2 tablespoons, then add the onion. Cook over medium heat, stirring, until the onion caramelizes, 10 to 15 minutes. Season with salt and cook until soft and evenly brown, about 3 more minutes. Transfer the onion to a plate and set aside.
Raise the heat level to medium high by moving the skillet so it's directly over the coals, or lower the grate so it's closer to the coals. Add enough canola oil to coat the skillet, about 2 tablespoons. Add the pastrami and cook, turning a few times, until crisp, 7 to 8 minutes. Remove the pastrami and set aside, leaving any accumulated juices in the skillet.
Divide the ground beef into 4 portions and form into 4 tight balls. Sprinkle them all over with salt and pepper. Place the hamburger balls in the skillet and sear 30 to 40 seconds, then use a strong flat metal spatula to flatten them. Continue to press with the spatula and cook until the burgers develop a crust on the bottom, about 2 1/2 minutes. Flip and cook to develop a crust on the second side, 1 minute. Use a spoon to baste the burgers with the pastrami fat in the pan.
Top each burger with 2 slices cheese. Pour 1 tablespoon water into the skillet and cover with a lid, aluminum foil or another pan to seal in the steam (this will help melt the cheese). Leave covered 30 to 40 seconds, until the cheese has melted, then remove the lid. When the burgers are at the desired point of doneness and the cheese is melted, top each with a pile of crispy pastrami, then set the burgers aside.
Brush the cut sides of the buns lightly with the garlic butter. Toast the buns directly on the grill over the campfire until golden and crisp around the edges, 15 to 20 seconds. Flip and lightly toast the outside of the buns, another 15 seconds.
Assemble each burger by layering the bottom half of each bun with pickle slices and caramelized onion, then a pastrami-burger patty. Top each with lettuce, tomatoes and the top half of the bun. Serve with caraway coleslaw.
Photograph from "Guy on Fire: 130 Recipes for Adventures in Outdoor Cooking"