Combine the oil, juice, garlic and rosemary in a small bowl. Remove the stem from each mushroom. Brush the mushrooms all over with the oil mixture and set on a large plate. Season mushrooms with salt and pepper. Let mushrooms sit at room temperature to absorb those flavours for 30 minutes.
Preheat an indoor or barbecue to medium, medium-high. Grill the mushrooms, starting cap-side down, 2 to 3 minutes per side, or until tender. Top the mushrooms with cheese and cook 1 minute more, or until it melts. Spread the buns with the mayonnaise.
Divide and top bottom buns with spinach. Set on the mushrooms, tomato, onion and top buns and serve.