Make a vinaigrette: Whisk 2 tablespoons of red wine vinegar with the mustard and some salt in a bowl. Drizzle in 1/3 cup of olive oil, a little at a time, whisking. Season with the herbs, black pepper and more salt to taste.

With a spoon, scrape and discard the dark gills from inside the mushrooms. Pour about a tablespoon of vinaigrette into the open bottom of each mushroom and set aside to marinate.

Drop the diced potato into a pot of boiling salted water and cook until almost tender, about 5 minutes. Remove with a slotted spoon. Add the chard leaves to the pot and cook until the leaves are tender, 4 to 5 minutes. Drain and chop coarsely.

Heat 2 tablespoons of oil in a large skillet. Cook and stir the reserved chard stems and the red onion until the onion is translucent.

Stir in the garlic and the pepper flakes and cook and stir a few more minutes. Add the cooked diced potato along with the chard leaves and about 1/3 cup of water. Cook until the water is absorbed and the vegetables are heated through. Season with salt and 2 teaspoons of red wine vinegar.

Heat the grill. Grill the mushrooms on both sides until they begin to brown and are soft to touch, 5 to 8 minutes.

Divide the filling into the mushroom bottoms. Sprinkle each with about a tablespoon of the cheese and grill until the cheese melts. Serve immediately.