Preheat a grill or grill pan to medium-high.
In a mixing bowl combine the olive oil, balsamic vinegar, herbs, garlic, salt and pepper and whisk to combine. Lightly brush the zucchini slices on both sides with the olive oil mixture. Place the zucchini slices on the grill and cook for 3 to 4 minutes per side, or until the slices are crisp-tender with nicely browned grill marks. Transfer to a serving platter and sprinkle with the kosher salt, if using. Repeat with the remaining vegetables, fitting as many on your grill as possible, and cooking until crisp tender. Garnish with Parmigiano-Reggiano, if desired. Serve hot or warm.