In a large resealable bag or shallow dish, mix all the marinade ingredients. Add the mushrooms and gently toss to coat. Marinate for 10 minutes, flipping once.
Heat the grill or grill pan to medium high heat. Make sure your grill grates are clean. Oil the grates right before cooking.
In a medium bowl, mix the fresh mozzarella, tomatoes, 1 tablespoon of basil and vinegar. Season with salt and pepper to taste.
Remove the mushrooms from the marinade and place them on the hot grill, stems side down. Cook for about 4-5 minutes. Flip the mushrooms and cook on the other side for about 5 to 6 minutes or until the mushrooms are heated through and tender.
To fill the mushrooms with ease, you may want to remove the mushrooms from the grill and place them on a plate or sheet pan. Fill each portobello mushroom with the caprese salad and close the grill lid to allow the heat to melt the cheese. If using a grill pan, cover the grill pan with a large lid (from a pan/pot) or place the grill pan under a pre-heated broiler for a few minutes or until the cheese melts.
Remove from the grill (or broiler), Drizzle with balsamic glaze and garnish with the remaining 1 tablespoon of basil. Serve immediately.
Notes
Remove the mushroom's gills: Portobello gills are edible, but they can darken the filling and release extra moisture. Removing them makes more room for the mozzarella and tomatoes.
Don't over-marinate. Portobello mushrooms absorb flavor quickly. Ten minutes is enough, and 30 minutes is plenty.
Oil the grates right before grilling: Mushrooms can stick, especially once they soften. Clean, hot, lightly oiled grates help prevent tearing.
Start stem side down: This helps the mushroom caps soften and begin cooking before you flip and fill them.
Mozzarella: You can use water-packed fresh mozzarella drained, cut into 1/2-inch cubes or shredded mozzarella.
Tomatoes: Instead of cherry tomatoes, you can also use 2 regular tomatoes, chopped into small chunks.
Make your own balsamic glaze Use approximately 1/2 cup of balsamic vinegar. This will yield about 1/4 cup of glaze or less depending on how thick you want the glaze. Simply pour it into a small saucepan and bring to a boil. Reduce until syrupy.