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Leeks are like the Lord Thistelwick Flanders of the onion family. The refined and aloof European cousin who needs to be nudged before his true onion character emerges. But once you start cooking with them, they offer a variety of characteristics that you don't find in regular onions.
First off, texture. When you cook down a regular onion, it becomes very soft, or it completely disintegrates. Leeks, on the other hand, retain a bit of tender bite, acquiring a pleasing texture more similar to that of, say, cooked cabbage. This property makes them fantastic whenever you want to add nice mild onion flavor without the stringiness or pulpiness of regular onions. Cooked very slowly in emulsified butter (often described on menus as "leek fondue"), they are incredible.
Flavorwise, they are far more mild than a regular onion, without the sweetness of a shallot. Slightly grassy and vegetal, they are known as the "soup onion" for a reason—they are great in soups and stews. Their flavor tends to sit back and support other flavors without overwhelming them.