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Whisk 2 tablespoons oil, vinegar and garlic in a bowl. Brush the mixture all over mushrooms and let stand for 30 minutes.
Meanwhile, heat the remaining 1 tablespoon oil in a large skillet over high heat. Add onion and cook, stirring frequently, until beginning to brown, about 5 minutes. Reduce heat to low, add water and cook, stirring frequently, until the onion is browned and very soft, about 15 minutes. Add port; cook, stirring occasionally, until mostly evaporated, about 3 minutes more. Stir in 1/4 teaspoon salt and 1/8 teaspoon pepper. Remove from the heat and cover.
Preheat grill to medium.
Sprinkle the mushrooms with the remaining 1/4 teaspoon salt and 1/8 teaspoon pepper. Grill, gill-side down, for 5 minutes. Turn over and top each with 2 tablespoons cheese. Grill until the mushrooms are tender, 4 to 5 minutes more.
Toast buns. Divide the onions among the mushrooms. Serve (cheese-side up) on buns with arugula and tomato.