Combine the mayonnaise, garlic, and rosemary in a small bowl. Set this aside to allow the garlic to mellow into the mayonnaise while you prepare the mushrooms.

Brush both sides of the mushroom halves with half of the oil and season with salt and pepper. Heat a large nonstick grill pan over medium-high heat until the oil shimmers but does not smoke.

Arrange the mushrooms in the pan and place a heavy Dutch oven or large cast-iron skillet directly on top to weight them down. Cook until the mushrooms are tender and charred, about 4 to 6 minutes per side—they should look flattened and have deep grill marks. Transfer to a plate and wipe the pan clean.

Spread the herb mayonnaise onto one side of each bread slice. On four of the slices, layer a quarter of the cheese, the cooked mushrooms, and the roasted peppers. Evenly distribute the remaining cheese over the peppers and top with the second bread slice, mayo-side down.

Brush the outside of the sandwiches with the remaining oil. Return the grill pan to medium heat and add the sandwiches. Weight them down with the Dutch oven and toast until the bread is deeply browned and the cheese is fully melted, about 3 to 5 minutes per side.

Serve immediately while the bread is still crisp and the cheese is molten.