Trim the radicchio heads, discarding wilted or bruised outer leaves and slicing off the very bottom if tough and discolored (don’t remove the core). Slice the heads in quarters or sixths, into wedges about 3 inches wide. Cut through the core, so the leaves are held together. Arrange a rack in the center of the oven, and heat to 375°F.
Put the butter and 2 tablespoons of the olive oil in the skillet, and set over moderate heat. When the butter is melted and foaming, stir in the onion and garlic and cook for a minute, then lay in all the radicchio wedges in one layer. Sprinkle on the salt, cover the pan, and cook slowly, turning the wedges over and stirring the onion every couple of minutes.
After 10 minutes or so, when the radicchio is softened slightly but still firmly holding its shape, stir the vinegar with ½ cup water and pour into the pan. Raise the heat a bit and bring the liquid to the boil, turning the wedges and stirring. Cook for a couple more minutes, until the pan juices are reduced and syrupy and the wedges are lightly caramelized.
Remove the skillet from the heat, and arrange the radicchio wedges in the baking dish in one layer. Spoon the onion all around, and pour the skillet liquid over. Drizzle on the remaining 2 tablespoons olive oil, season with freshly ground black pepper, and sprinkle the grated cheese in an even layer, covering the radicchio.
Tent the dish with aluminum foil, pressing it against the sides. Bake covered for about 20 minutes, remove the foil, and bake another 5 minutes or more, until the radicchio wedges are tender and moist and glazed golden on top.