Bring a large pot of water to a boil, season with salt, and add potatoes. Reduce heat to a simmer, cover, and cook potatoes until tender when pierced with a knife, about 15 minutes. Drain potatoes and let cool.
Meanwhile, cut radicchio into quarters and remove cores. Tear radicchio into bite-size pieces.
Combine vinegar, salt, pepper, mustard, garlic, and oil in a jar. Shake to mix well. Taste and add more salt and pepper if you like.
Crush potatoes slightly with bottom of a heavy glass or a potato masher (this helps them absorb vinaigrette). Combine potatoes, radicchio, and chickpeas in a very large bowl (or 2 bowls). Toss salad with vinaigrette and arrange on platters.