Heat a large skillet over medium heat, add truffle oil and heat through. Add the mushrooms and sauté, stirring occasionally, about 3 minutes. Season with salt so the mushrooms release their moisture and they begin to shrink. Increase the heat to high and continue to sauté until the mushrooms are golden brown, about an additional 3 minutes. Season with pepper then remove from the pan to a small bowl and set aside.
Heat a large skillet or griddle over medium-low heat. Spread one side of each bread slice with mayonnaise. Place half of the slices of bread, mayo-side down, on the pan or griddle. Divide the cheese evenly on top of the bread slices. Then divide the mushrooms and arugula on top of the cheese. Top the sandwiches with the second slice of bread, mayo side up.
When the underside is golden brown, about 4 minutes, use a spatula to flip the sandwiches over. Gently press down on the sandwiches to help melt the cheese but be careful not to smash it. Cook until the second side is golden brown and the cheese is melted.
Cut both sandwiches in half and serve immediately.