1. Whisk mayonnaise, garlic and rosemary together in a bowl. In a second bowl, toss mushrooms, oil, 1/4 teaspoon salt and 1/2 teaspoon pepper. Heat large nonstick grill pan over medium heat for one minute. Arrange mushrooms on pan, weight with Dutch oven, and cook until browned, about six minutes on each side. (Alternatively, used a panini press.) Transfer to a plate.
2. Spread mayo mixture evenly on 1 side of each slice of bread. Layer half the cheese on the mayo side of 4 bread slices, then top with mushrooms, peppers and remaining cheese. Arrange remaining bread slices, mayo side down, over cheese.
3. Return grill pan to medium heat. Place 2 sandwiches in pan and weight with Dutch oven. Cook until golden brown and cheese is melted, 4 to 6 minutes on each side. Transfer to wire rack and repeat with remaining sandwiches. Serve warm.
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