Halve each radicchio. Then thickly slice each radicchio into circles or cut each half into 4 sections, trimming and discarding the base and a little of the white heart from each section.
Halve the onions and trim the base of each. Cut each half lengthwise into crescent moon-shaped wedges, each about 3/4 inches wide at its thickest part. (Or just slice them horizontally, if you prefer.)
Mix the olive oil, balsamic, and seasoning together in a dish. Put the onions in a bowl and the radicchio in another bowl then drizzle both with dressing. Gently turn to coat and let sit for about 10 minutes.
Heat a skillet over medium-high heat until really hot but not smoking. Cook the onions quickly until well colored on all sides, turning once or more. Reduce the heat to low and let the onions cook, turning frequently, until softened, maybe 2 minutes or so.
Meanwhile, spoon the white bean puree into a serving dish, reheating it gently first if you want to. The puree can also be served at room temperature.
Increase the heat under the onions to medium-high, add the radicchio to the skillet, and cook just until the radicchio is wilted and colored on each side—this will happen very quickly, say, 45 seconds or so. Pile the radicchio and onions on top of the white bean puree. Season with salt and black pepper, and serve with a little oil drizzled over the top.