01 - Build a steady campfire with hot coals or preheat a grill to medium-high heat, approximately 375°F to 400°F.
02 - In a large bowl, combine the ground beef with salt, pepper, garlic powder, and smoked paprika. Mix gently until evenly seasoned, then divide into 4 equal portions and shape into flat, even patties.
03 - Tear 4 large sheets of heavy-duty aluminum foil, approximately 18 inches each. Lightly drizzle the center of each sheet with olive oil to prevent sticking.
04 - Arrange thinly sliced potatoes in the center of each foil sheet. Top with layers of red onion, bell pepper, mushrooms, and halved cherry tomatoes. Season the vegetables with salt and pepper.
05 - Place one seasoned beef patty on top of each vegetable pile. Drizzle each patty with ketchup and yellow mustard.
06 - Fold the foil tightly around the contents, sealing all edges securely to form enclosed packets that trap steam and juices during cooking.
07 - Place the sealed packets directly on hot coals or grill grates. Cook for 25 to 30 minutes, flipping once halfway through, until the potatoes are tender and the beef is cooked to an internal temperature of 160°F.
08 - Carefully open each packet, watching for hot steam. Place a slice of cheddar cheese on each patty, reseal the foil, and let rest for 1 to 2 minutes until the cheese melts.
09 - Serve the packets hot directly from the foil, or transfer the contents onto hamburger buns with additional condiments as desired.