Oh, celery, how much do I love thee? Let me count the ways... chopped, shredded, boiled... well, three ways then.
Making stock is an excellent use for all leftover vegetables, whether they're whole and unused or trimmings from some other recipe. I've taken to an anachronism of late, freezing the results rather than keeping a stock pot going, despite nobody - almost literally - owning a deep freeze in the 1940s in the UK. But it is possible to keep a stock pot running without the freezer.
The essence of a stock is basically just raw stuff boiled. It's probably easier to use cubes, but they're usually more salt than anything else. Homemade stock doesn't need salt in advance, allowing you to regulate the amount in the cooking itself. CJBS, at 58, watches his salt intake like it may turn on him at any time, despite having normal levels of cholesterol in his system. At 35, I'm less good at this, despite having hypertension and swallowing bucketloads of statins every day (oh, the joy of having a thyroid gland that no longer works).
A "live" stock pot, with no freezing, needs managing. You start it off with the trimmings of any root vegetables you've got, including the skins. Onion skins are brilliant here, giving the stock a lovely golden colour. Put the trimmings in a big pot, pour on boiling water to cover and bring back to the boil with the lid on. If you bring this back to the boil - vigorously - every day, you can keep adding vegetable matter to it and topping it up with more water each time you use it. The Ministry of Food encourage everybody to do this during the war. In theory, you can get a month out of this before it starts to go bad despite the boiling (it starts to smell sweet, and as soon as this happens, throw the entire lot away and throughly wash the stock pot). I've found that actually three weeks is more likely than a month.
If you add green vegetables, reduce the amount it will last by more than a half. It'll taste even better with green or leafy vegetables, but it simply won't survive as well.
Adding the bones of any meat you've had also reduces the amount it lasts, and adds the issue of the unattractive globs of fat on the surface. There are three ways to deal with this: don't get them in the ladle when you decant the stock; warm the stock and attempt to dab the top with strong kitchen paper (this never works very well), or decant the stock into a bowl through a muslin towel (this is more trouble than it's worth).
A stronger stock is got by the repeat-boiling method, and if you're using the deep freeze, it's still worth keeping the pot going for a couple of days at least before decanting it. I label up the stock with the dominant ingredient (judged by smell) and freeze it in useful-sized amounts. The results go into soups, stews, curries, gravy - anything that I'd put a stock cube into if they were freely available to me (they're a points item, so I need to be careful with them).
If you're waste-neutral, the multi-boiled trimmings at the end compost very very well. Any red meat bones you may have will also go very well with your own or a local dog, judging by our two. Poultry bones can't go to dogs, but a cat will secretly plan to murder you in the night to get hold of them, if our late, but uniquely wonderful, cat Freddie was anything to go by.
Having got hold of some (not very fresh) celery at my not-very-local Co-op branch (I had £12 in dividend to spend, making it worth the train trip) I used plenty in recipes and chucked the rest into my latest stock pot. It's just had its second of three boils, then I'll freeze it against the day when the local supermarket goes back to selling higher-margin celery flown in from Spain, Morocco and Egypt.
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