To come straight to an answer which may be of relevance to you, consider that the onion family has become adapted to a wide variety of growing environments. One member of that family is the Egyptian, Tree or Walking onion which develops clusters of small bulbils at the end of the flowering stalk. These become heavy, flop over and take root at a small distance from the mother plant. So a cultivator can take advantage of this behaviour by removing a mature set of bulbils, separating them to individuals and planting each one favourably to produce a single large bulb. Since they are in this sense perennial, this fits your description of plant, harvest, plant, never actually going through a seed/seedling stage.
More generally onions are biennials which from seed build a store of resources in the first year, which is followed in the second year by the flowering and seeding stage. Since it can take some time to build the required resources the plants have adapted to taking one full season for the first stage, implementing a shut down to survive the winter or rest period and then reawaken to produce the reproductive behaviour. In some seasons the trigger to move to reproduction can happen early, particularly when the plant is under stress, perhaps due to drought or unusually long spells of sunshine.
Since the onion family is adept at adaptation, there is also the planting of setts, or onions grown from seed to a small size and then artificially forced into early maturity. These are then planted the following spring and allowed to complete their normal first year activity by expanding the small bulb to a large store of resources that is useful as food.
