Place a baking rack over a large rimmed baking sheet. Clean the portobella mushroom caps by gently wiping them with a damp paper towel. Then remove and discard the stems. Place the mushroom caps gill side down on the baking rack. Set aside.
In a small bowl, combine the olive oil, balsamic vinegar, salt and pepper. Brush some of the dressing over the surface of mushroom caps. Transfer the mushroom caps to the oven and bake for 15-18 minutes. The mushrooms should be soft, but still firm enough so that they hold their shape. Remove the mushrooms from the oven and set aside to cool slightly.
Meanwhile, in a medium bowl gently toss together the arugula, strawberries or tomatoes, mozzarella and basil with the remaining dressing.
Using a pair of tongs, flip the mushroom caps over. Fill each mushroom cap with some of the salad mixture. Serve immediately.