Does it matter how you cut garlic?
Garlic contains volatile oils that are released when it's crushed or pressed. As garlic's cell walls are smashed, its oils react with its natural enzymes, and the aroma and flavour become very strong. If used immediately in raw preparations, the pungent pulp and extracted juices from pressed or pureed garlic give your dish a pronounced spicy flavour. Unfortunately, these oils don't last but turn rancid quickly and linger on hands, breath, and cutting surfaces. Pressed garlic doesn't hold up well when heated, either. It turns bitter and quickly loses its characteristic garlic flavour. I would advise against using a garlic press for these reasons. Plus, they are difficult to clean.
When you mince or chop garlic, the oils aren't violently forced out but are left to slowly season your food as its cooks. Also, the enzymes that make garlic pungent are destroyed by heat, so the garlic flavor is more apt to mellow as it cooks.
In general, the longer the cooking time, the larger you can leave the pieces of garlic. Finely minced garlic may also be used in vinaigrettes and salsas where the high acidity of the food will help break down the garlic and bring out its flavours.
The more aggressively garlic is handled, the more aggressive and short-lived its flavour.
As I understand it, the main reason for adding salt to the garlic when dicing or pressing it, it to absorb the precious oils. This means that more of the oil ends up in the pot, and not on the chopping board, and not on your fingers.
Once you have peeled the garlic (in itself quite an art, see Footprints' writeup below), you will be ready to start the preparation. My best experiences have been when I have roughly chopped the cloves, then gently pressed them with a broad-bladed knife (I use Sabatier). Sprinkled with just enough salt to absorb the juices, the pulp is then gently mixed on the board then transferred to the pot. The juices mingle straight away, and the larger chunks marinate in the mix, slowly softening and releasing their goodness.
I have also tried adding a little more, finely-chopped, toward the end of the cooking process. The freshness of the new garlic has to be tasted to be believed. This way, I get the best of all worlds when cooking. Word of warning though - when adding garlic to salads, you may need to do a final chop before adding it, as many people find larger pieces of garlic too challenging!
Take a clove of garlic, and start peeling it with your hands. Become very frustrated. Peel layer after layer after layer. Work really hard to get that last layer off. Watch your fingers slip off the garlic peel, and listen to yourself curse with vigor and gusto.
Advantages The only method where the garlic comes out complete and intact.
Disadvantages Very smelly fingers. Slow. Bad-temper inducing.
Grading
Advantages Quick and enjoyable. Very little smell stays on the fingers, due to minimal contact with the garlic. It is definitely the most fun way of peeling garlic. (Not in the grading system, admittedly, but should definitely be noted). It is a good method to impress people with, as violent means to non-violent ends are often amusing.
Disadvantages You get a pulp of a clove, so this is good only if you want crushed garlic, or have some other amorphous use for garlic.
Advantages I've seen several chefs teach this, so it must be good.
Disadvantages More garlic handling than 'crushing' and 'the plastic tube', especially if you don't get it right the first time. It's a bit harder to master than the others.
Advantages Your hands never come into contact with the garlic, so there is no smell. It is very simple to use, and basically does a great job.
Disadvantages You have to buy a tube, and you may end up squishing the garlic slightly.
Regarding the smell - Teiresias says "Have you tried washing your hands with a little milk?". Also, there is an interesting idea involving stainless steel in the above writeups.
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