- Slice off 1/2" of the top of the onion (opposite the root)
- Place onion root side up, and make slices all the way through, staying at least 1/2" away from the root
- Flip it over and spread apart the leaves
This is best described with images.
Alternatively, if you're willing to drop a few hundred dollars, you can buy a bloomin' onion cutter. This is what restaurants use. There are cheaper alternatives, but I can't vouch for their quality or ease of use.
I've never tried to store them myself, but one of the books that I got when I started gardening was the "Encyclopedia of Country Living", which is loaded with information on growing, canning, storing, etc. Its recommendations for globe onions is a follows (minus the typos of my transcribing):
Curing Bulbs. Let them sun-cure for 3 days. Spread them out on top of the ground until tops are thoroughly dry. Then sort. Bulbs that addicentially got cut with the space go into the house to be used soon. Sets that for some reason didn't take off are cured some more (maybe 2 weeks in a shady, airy place) and then saved to be given another chance next year. They often make it the second time they are planted. The winter storage onions are cured some more and then bagged for keeping.
Storing. If properly cured and stored in a dry, airy place, many of your winter-keeper onions will last 4 or even 6 months. Every summer we bring in onions from sets by the gunnysackful for our winter supply. The trick is to sure them very well in the sun for 3 days or so before bringing in. Rip off the "set" portion before drying if you can. The onion grown from a set has 2 parts. One is small and extends up into the a stiff hollow stem. That is the part you should rip off -- or else the onions won't keep long.
After you bring your onions in from the field, continue drying them. We have tried all sorts of systems for this. You can braid them by the dry stems and hang them up in bunches with wire or twine around the bunch. Or them them in an "onion"-type bag or into old pantyhose and hang them up. My current method, and easiest one yet, is to dump them into cardboard boxes and bring them into the kitchen. I check them occassionally by running my hands through the boxes and removing any damp ones. Any onion that feels the least bit damp is on the verge of spoiling. When the rest are absolutely bone-dry from the kitchen's heat (once that old wood cookstove starts going, it really gives them the treatement), they can go under the bed or into the attic -- anywhere they won't freeze and will stay dry.
Slight freezing doesn't hurt them, provided you don't handle them while frozen. Good ventilation is very important to their storage place, and what's why they should be in some sort of loosely woven bag or hanging basket.or braided and hung up or some such, although I get away with a box. They do better stored dry and cool than dry and hot because with too much heat, they have a tendancy to gradually dry away to layer upon layer of papery nothingness as winter goes by.
When we use our big winter onions, we take the largest first and gradually work our way down in size. That way, if there are any left over come spring, they will be small ones suitable to use as sets in the spring planting.
I should also note that I've seen the pantyhose method in use before, I think at my godparent's place. They had loaded an onion into the toe of the stocking, tied a knot, loaded in the next one, tied a knot, etc, so the onions didn't actually touch which might affect their drying out, and then hung them up so they weren't touching anything.
Best Answer
When I use fresh onions, we store the unused parts for up to a week in our fridge in either a ziploc or a sealable rubbermaid-style container. For particularly pungent onions they go in the crisper drawer to keep the smell from being too strong in the rest of the fridge, but usually they're just on one of the shelves. Stilltasty says 2-3 days but my personal experience says otherwise.
Generally in our house, they don't last long enough for the flavor to degrade too much, but they will get a little dried and the smell will get stronger over time. After a week, they shouldn't be bad in the sense that you'll get sick - but you'll want to use them in a meal where the onion is the star, as the aroma will take over the dish.
Another option that I'm a big fan of is to dice or strip the leftovers (or, a 5 lb bag at once!) and freeze them on a cookie sheet (instead of in a bag, where they'll turn into a giant clump). Once fully frozen, dump them into a plastic freezer bag, and store in the freezer for easy access to a handful at a time. For most applications, you can defrost them right in the pan/pot/casserole as you cook, so they're very handy and accessible. We do the same with peppers in our house for convenience.