Oberothorn which is 3400+m is supposed to be the highest hiking peak in Europe, start from
Fluhalp or Zermatt. I think it will be sufficiently snow-free for it to be called "hikeable" only for a few weeks of the year, this weekend could well be the last of the season, so hurry up, and check before you go.
And would be good if you come back and write a short update, as there might be others interested,
If you ask me, off the record I would say the hike from Testa Grigia hut (3500m, Italian side) hut to Klein Matterhorn (3800m) is much higher; no crevasses if you stay close to the piste edge, but the speeding torpedos/out of control skiers pose a serious danger; officially it is not a hiking path but unofficially it is what everyone uses to enter CH from Italian side.
You can also walk from Zermatt to Testa Grigia but officially it goes over a glacier (Theodulpass) and mountain guide/roping up is recommended. Unofficially you can consult the hut staff and start early, it should be safe enough, but conditions change ever so often,
so check locally before you go. The whole route from Gandegg hut (3000m) to Kl Matterhorn (3800m) is snow covered and fun, straightforward walk.