Shallots like the other multiplier onions produce a "nest" of several bulbs, each distinct although connected at the root, and are ready to harvest like onions, when the leaves fall over and go brown. The nests should be right at the surface and easily found even after the leaves have pretty much dried off. They are harvested just like onions and cured the same way.

Most of the shallots, except for the French grays, are true onions, they just divide into a number of bulbs for each one planted, and seldom seed, at least the ones you plant from seed bulbs in the fall or early spring. One whole part of the true onion species does that, with shallots only a part of that segment. The French grays are a species all to themselves.

The ones you grow from seed like regular onions tend to produce one bulb per seed, though, at least in the first year. How well they do when planted back in a traditional manner in following years is something I have no experience with. Shallot seed is a relatively recent development.

FWIW the term "shallot" is used in a number of different ways depending on what part of the world one is considering. There are also Dutch shallots and French shallots beside the already mentioned grays, and both of them come in yellow and red varieties with the line blurred badly between shallots and potato onions, the later also coming in white. See Allium cepa var. aggregatum