Prepare a charcoal grill to get a medium-hot fire or set a gas grill to medium high.
Season the onion slices with olive oil and salt. The easiest way to do this is to coat the bottom of a platter or baking sheet with olive oil and sprinkle salt on top -- just as if you were seasoning the sheet pan. Carefully set the onion rounds in the oil, aiming to leave them intact so that it's easier to grill them. Sprinkle the top of the onions with salt and drizzle with a touch more olive oil. To season the radicchio, toss the wedges in a large bowl with a generous sprinkling of salt, about 3 tablespoons of the olive oil, and the balsamic vinegar.
Grill the onions and radicchio over the medium-hot fire until tender. It's usually easiest to grill the onions first; brown them on both sides and then scoop them into a pile and transfer them to a cooler spot on the grill. Let them finish cooking slowly while you grill the radicchio directly over the medium hot fire. The radicchio is done when it's wilted and browned all over.
Roughly chop the onions and radicchio and toss them together in a bowl. Taste for seasoning and add salt, pepper, and more balsamic vinegar, if needed. Top each slice of grilled bread with the radicchio and onions and finish with a fresh drizzle of olive oil, a few twists of pepper, and a few drops of balsamic vinegar.