Preheat oven to 450 degrees. Arrange mushrooms on a baking sheet, stem sides down, and brush with 1 tablespoon oil. Season with 1/4 teaspoon salt and some pepper. Toss tomatoes with remaining tablespoon oil, the sliced garlic, and crushed red-pepper flakes on a parchment-lined rimmed baking sheet.
Roast, flipping mushrooms and stirring tomatoes halfway through, for 25 minutes. Transfer tomatoes to a bowl, and toss with reserved juice.
Heat broiler. Toast buns, cut sides up, on a baking sheet for about 45 seconds. Broil mushrooms on center rack for 3 minutes. Flip, and top 4 with cheese, then remaining mushrooms. Broil just until cheese melts, about 1 minute.
Sandwich tomatoes, arugula, and mushrooms between buns.