Preheat the grill to medium-high heat. Prepare the portobellos by removing the stem and scraping off the gills with a spoon. Brush each side of the caps with half of the olive oil, and season with salt and pepper.
Place the portobello caps directly on the grill racks or on grill mats. Cook for 3 to 5 minutes on each side until tender.
While the caps are cooking, prepare the caprese salad by mixing together the cherry tomatoes, chopped mozzarella, basil, the other tablespoon of the olive oil, and the balsamic vinegar. Add salt and pepper to taste.
Next, stuff the mushrooms. If you want the caprese filling to be warm and the cheese to melt, stuff the mushrooms while they are on the grill. This doesn't work as well if the caps lost some of their shape while cooking. If you do it this way, you may also want to slice the tomatoes into smaller pieces so they don't roll off the top while the mushrooms are cooking. Close the grill and cook for 2 to 3 minutes to warm the filling/melt the cheese. Remove and transfer to a serving dish.
Alternatively, you can remove the portobellos from the grill and transfer them to a serving dish before stuffing with the caprese salad. Either option is delicious, but the latter method is usually easier/less messy.
Drizzle the mushroom caps with balsamic reduction (if you have it) or more balsamic vinegar. Add flaky sea salt, ground black pepper, and more fresh basil to taste.