Preheat a grill for direct heat cooking and preheat the broiler. Rub the mushroom caps with the olive oil and season with salt and pepper. Place on the grill and cook, turning once, until softened, about 5 minutes. Cool.
Divide the beef into 4 equal amounts and working with one portion at a time, form into a very thin patty. Place the mushroom cap in the center and fold the beef up and around the mushroom pressing the beef around the edges of the cap until the mushroom is completely covered. Season the mushroom-beef patties generously with salt and pepper and place on the grill. Grill 4 to 5 minutes on each side for medium rare.
Meanwhile, cook the pancetta in a medium nonstick skillet over medium-high heat until crispy and transfer to a paper towel-lined plate to drain. Generously spread the butter on both sides of each roll and place under the broiler until toasted.
Place the cooked burgers on a rimmed cookie sheet and evenly top with the crispy pancetta bits and a slice of Swiss cheese. Place under the broiler until the cheese is just melted.
Build the burger by putting the patty on the bottom of a toasted bun. Add your favorite toppings and condiments then add the top half of the bun. Enjoy.
NotesBy putting the mushroom inside the beef patty, not in chunks mind you, but a whole portabella mushroom cap, I ensure you get a good bit of mushroom in every single bite. Instead of topping the burger with soggy strips of bacon, I use crispy lardons of pancetta then cover the whole thing with aged Swiss cheese on a properly griddled bun for the Perfect Bacon-Mushroom Burger.