Cut off the stem inside the portobellos. With a spoon, scrape out and discard the dark gills.

In a shallow bowl make a marinade with the vinegar, olive oil, thyme and plenty of salt and pepper. Add the mushrooms, turning them occasionally so all sides are flavored.

Pick through the arugula and spinach, striping away and discarding the tougher stems. Chop coarsely.

Heat a grill to medium high heat. Grill the marinated portobellos about 5 minutes on each side, until tender. Be sure to save the leftover marinade. When the portobellos are tender, transfer them to a plate, concave side up.

Meanwhile prepare the filling. Heat some oil in a large skillet on medium flame. Add the onion and sauté until translucent, about 5 minutes, then add the garlic and cook another minute. Stir in the chopped greens and cook until they wilt. Add the leftover mushroom marinade and continue to cook and stir until the liquid evaporates. Season to taste with salt and pepper.

Divide the spinach and arugula filling among the grilled mushrooms. Arrange mozzarella on top, more or less, as you prefer. Return to the grill and cook about 10 minutes, until the cheese melts over the filling.

Serve with tomato sauce, if desired.