Heat a gas grill to medium or prepare a medium charcoal fire. (Be sure the grate is hot, too.) While the grill is heating, trim the ends off the onions and slice the onions into 3/4-inch-thick rounds. Spear each round with a skewer through one side and straight through the center to keep the onion rings intact (each skewered onion will resemble a lollipop). Break off excess lengths of the wooden skewers, leaving about 2 inches exposed for removing the skewers after grilling. Brush the onions with olive oil. Season with salt and set aside.
When the grill is hot, arrange the onion skewers on the grate directly over the heat and grill, turning every few minutes as needed, until the onions are soft and well charred, about 20 minutes total. Remove from the grill, take the onions off the skewers, discard any crisp, blackened outer rings, and wrap the onions in foil. Set aside for 10 minutes.
Meanwhile, lightly brush olive oil on both sides of each slice of bread and lightly sprinkle with salt. Put the bread directly on the cooking grate and grill on both sides to toast, about 2 minutes per side. Immediately after grilling, rub with the cut side of the garlic on both sides. Set aside.
In a small bowl, whisk the olive oil with the balsamic vinegar and season with salt and pepper to taste. Unwrap the onions and put the rings in a medium bowl; toss with about half the balsamic vinaigrette. To serve on individual plates, put a slice of toast on each of four plates and drizzle with the remaining dressing. Top each slice with an equal amount of the charred onions. Drizzle with a little olive oil, if you like. Sprinkle on the parsley, prosciutto, and Parmesan shavings. Season with freshly ground pepper and serve.
To serve on a platter, cut the bread into crouton-size cubes, toss with the dressed charred onions and the remaining dressing, season with pepper to taste, and top with the parsley, prosciutto, and Parmesan shavings.