What you may not know about onions is that they have a season, like everything else. Those sharp, pungent specimens you buy at the grocery store are called “storage” onions. They're grown until the outer skin becomes leathery or papery (depending on the variety) so that they can be kept and shipped through the non-onion season, allowing French onion soup to be created throughout the year.
Fresh onions, however, are available only in late spring and early summer. Just like their cousin fresh garlic, they come to the market with the stem intact. They also come pre-peeled to show off their beauty. Choose onions with the freshest stalk and the perkiest-looking roots possible. While a gash in the outer layer is acceptable (you can just peel off the outer layer), avoid deep gouges; they may be indicative of problems deeper within.
The stem is edible, though often tough; a great use for the stems is tossed, leek-like, into a pot of slowly, slowly bubbling stock: chicken, beef, lamb, veal, even pork broth. Ecumez, dépouillez !–or your stock will be cloudy.
As for the bulb of the fresh onion, you can treat it like a storage onion, though it has a haunting sweetness hiding behind the bite that is lost somewhat in translation. One of my favorite things to do is to French the onion (see below) and then simply toss it with extra-virgin olive oil, salt and pepper. Let it sit for a little while to get the onion juice to puddle throughout the oil, toss again and eat as a salad.
Fresh red onions transform the traditional Yucatecan pickle, cebolla encurtida, into something worth eating on its own; French a fresh red onion, blanch them (dip them for thirty seconds into boiling water and evacuate to cold water), then add a teaspoon of peppercorns, a teaspoon of allspice, a tablespoon of dried oregano, a handful of fresh thyme, 2 cups of vinegar, the juice of two American limes (green lemons) and a big pinch of salt. Put in a jar, shake and let sit overnight.
Finally, the best part about sweet onions is that they make incredible sofregit, the “mother base” of Catalan cooking. Sofregit, which is related to the Italian and Spanish sofrito/sofritto, is a large pile of onions that has been cooked until completely wilted, with tomatoes or tomato paste added in the last several minutes of cooking. Just put olive oil, Frenched onions and salt into a heavy pot, then set over low to medium heat until just starting to caramelize. Use as a starter for everything from seafood soup to pasta sauce.
To French an onion, which refers to cutting, not kissing, first halve the onion from root to stem. Cut off the root and the stem and turn each half so the cut part is touching the board. With a long knife (a chef's knife, for example), cut radially around the onion toward the middle. The onion itself wants to help you here: the striations of an onion are much clearer on a fresh onion than on a dried onion, and just cut wherever you see a little “valley”. Once you've made it halfway up the onion, turn it around and cut radially up the other side.
When you're done, you'll have a pile of various-sized crescents that work wonderfully tossed into salads, are easy to sauté, or which create textural interest in a bowl of, yes, soupe à l'oignon.
If Frenching isn't your thing, you can always halve the onions, brush with olive oil, and grill until char marks are found on the bottom; dipped into mayonnaise or romesco sauce, they are a sweet treat.