Preheat the oven to 325 F. Put the potatoes in a large saucepan with cold salted water to cover and bring to a boil over high heat. Cook until tender, about 10 minutes. Drain well, scatter on a baking sheet, and let cool for several minutes. Place in the oven to dry out, about 5 minutes. Let cool, then press through a ricer, the large holes of a box grater, or a colander.

While the potatoes are in the oven, cut the onion half into quarters and slice thinly to yield slim pieces about 1 inch long. Put in a non-reactive bowl with the vinegar and oregano and let rest for 15 minutes.

Season the potatoes with salt and pepper, then divide into 8 equal balls. Place a ball in the palm of your hand, top with a piece of cheese, and then a second potato ball. Shape the potato around the cheese to form a 4-inch patty. Repeat with the remaining potato balls and cheese. (The recipe may be prepared to this point, covered, and refrigerated up to 1 day before cooking.)

Heat the 2 tablespoons olive oil in a nonstick skillet over medium heat or in an electric frying skillet set at 350 F. Add the polpette and cook, turning once, until brown and crispy on both sides, about 20 minutes total.

While the polpette are cooking, finish making the salad. Whisk the remaining 1/2 cup olive oil and salt and pepper to taste into the onion-vinegar mixture. Add the cucumber, avocados, and tomato and toss well.

Arrange a nest of lettuce on each of 4 plates. Spoon the salad over each nest. Top with a polpette and spoon more dressing over all. Serve at once.