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Sauteing is the process by which a vegetable is sliced and brought to a golden-brown by cooking it over a medium-low to medium heat with a small amount of butter or other fat. It is the first step in a great number of recipes.
The first rule of sauteing is to go slow. Make sure that your butter is melted or your oil is relatively hot before you add your vegetables. Gauging the temperature of the oil can be accomplished by holding your hand about three or four inches over the bottom of the pan - if you can feel the heat, it's probably hot enough. The smaller or thinner you slice or dice your vegetables, the more evenly they will cook.
On the average, it only takes between three and ten minutes to saute a normal portion for most recipes. The wide variation is based on the sugar content of the vegetable in question. Onions have a relatively high sugar content, so they saute quickly. Celery has almost no sugar, so it hardly sautes at all unless it's mixed with something else that will saute.
A neat little trick to help stave off rampaging hoards of hungry children is to saute a half an onion on a back burner. Even if you don't use them in any of the dishes, they are often a nice topping or condiment, and the smell of the onions cooking satisfies everyone's need to know that dinner is on its way.