Prepare a gas, charcoal, or wood fire for grilling.
Grill the bread a few inches from the heat until golden on both sides. Place the bread on a platter and immediately rub the garlic cloves over the top of the bread's surface. For really garlicky fettunta, rub hard and use both cloves of garlic. Drizzle a generous tablespoon or more of olive oil over each slice of bread.
Combine the prosciutto, cheese, and parsley in a small bowl. Sprinkle the mixture evenly over each slice of bread. Return the bread to the hot grill cheese side up, cover the grill with the lid, and cook the fettunta just long enough to melt the cheese and rewarm the toast a bit, about 1 minute. Serve hot.