Begin by preheating a large skillet over medium heat. Place the cut sides of the hamburger buns down into the skillet and toast for 2 to 3 minutes until lightly golden. Remove the buns from the skillet and set them aside on a plate.

Prepare an outdoor grill for medium heat. In a large saucepan, melt butter over medium heat directly on the grill. Add the sliced mushrooms and onions, seasoning with salt and freshly ground black pepper. Cook, stirring occasionally, for about 20 minutes, or until the mushrooms are lightly browned.

While the mushrooms and onions are cooking, combine the ground beef and onion soup mix in a bowl. Mix thoroughly with your hands, then shape the mixture into six even burger patties.

Lightly oil the grill grates. Grill the patties for approximately 5 minutes on one side, then flip them and continue grilling until they reach your desired level of doneness. To ensure they are fully cooked, use a meat thermometer, with the internal temperature reaching 180°F (82°C).

Once the burgers are cooked, remove them from the grill and allow them to rest for 5 minutes. Spread a tablespoon of mayonnaise on each bun, then add a patty to each. Top with the sautéed mushrooms and onions, dividing them equally between the burgers, and serve.