Preheat a grill to medium. In a large nonstick skillet, heat 2 teaspoons olive oil over medium heat. Add the mushrooms; season with pepper and cook, turning occasionally, until browned and tender, about 5 minutes. Add 2 tablespoons water and cook, turning the mushrooms once, until the water is evaporated. Transfer to a cutting board and let cool; reserve the skillet. Finely chop the mushrooms and transfer to a bowl.
Add the remaining 2 teaspoons olive oil and the onion to the reserved skillet. Season with pepper and cook over medium heat, stirring, until the onion is tender and golden, about 3 minutes; let cool. Add half of the onion and the ground beef to the mushrooms; season with salt and mix well. Form the mixture into a ball, transfer to a board and cut into 8 portions. Form each into 3- to 4-inch patties.
Transfer the remaining onion to a small bowl, then stir in the mayonnaise and tomato paste. Spread the sauce on both faces of the rolls and arrange open-faced on each of 4 plates. Divide half of the lettuce among the roll bottoms.
Add the patties to the grill and cook for about 2 minutes. Flip, divide the cheese among the patties, cover with foil and cook until melted, about 1 minute. Stack 2 patties on each roll bottom, top with the pickle slices and remaining lettuce and set the roll tops in place.