Preheat a grill to medium high (or preheat an oven to 425°F).
Remove the stems from the portabello mushrooms. Mince the garlic. In a small bowl, mix the garlic with the balsamic vinegar and olive oil. Place the mushrooms on a baking sheet. Using a brush, brush the front and back of each mushroom with the marinade. Sprinkle with kosher salt and pepper, and let stand for about 15 minutes.
Chop the mozzarella into 1/4 inch chunks. Finely dice the tomatoes; place them in a colander to drain and sprinkle with kosher salt. Thinly slice (chiffonade) the basil.
When the grill is ready, place the mushrooms on grill (or in a grill pan) and cook about 5 minutes per side, until juicy. (Alternative method: Roast the mushrooms in a baking dish in the oven for about 15 minutes at 425°F, turning once.)
Remove the mushrooms from the grill. Top with chopped tomatoes, mozzarella, basil, and a drizzle of olive oil.