4 large portobello mushrooms, stems and gills removed
1 large sweet onion, such as Vidalia, cut into 1/2 inch thick slices
1/3 cup extra virgin olive oil
1 tablespoon balsamic vinegar
1 tablespoon minced fresh thyme leaves
    sea salt to taste
    freshly ground black pepper, to taste
4   thick slices Gruyere cheese
4 large rolls, cut in half and toasted
    mayonnaise
1 bunch fresh arugula or watercress, stems discarded