Preheat a grill to medium. In a large bowl, mix together the lamb, feta, oregano, 1/2 teaspoon salt and 1/4 teaspoon pepper. Shape meat into four 3/4-inch-thick patties.
Brush olive oil all over the mushrooms, onion and bread slices. Season the patties and vegetables with salt and pepper and grill, covered, until the patties are just firm to the touch, about 3 to 4 minutes on each side. Grill the bread until toasted, 1 to 2 minutes on each side.
Divide the bread among 4 plates and top each with a mushroom cap, a lamb patty and some onion slices; drizzle with the vinegar.