Caramelizing onions
The key to getting onions to form a nice caramel-like residue is to slice them very thin, and not allow moisture to build up in the pan while they cook at a moderate temperature ('7' on most range dials is good).
I prefer to cut an onion in half on the north-south axis (through the root and the shoot), not at the equator. Then I place each half flat-side down, and cut perpendicular to the first cut but still parallel to the north-south axis. Slice them as paper-thin as you possibly can...a very sharp chef's knife is a great help here. As you get toward the end of slicing one of these onion halves, it might become difficult to keep your grip on the remaining uncut onion. I find that if I start to angle my cuts with the top of the knife toward the remaining wedge, the wedge remains more stable because its larger part is against the cutting board.
Most of these extra-super-thin slices will stay together on their way to the pan. Use both hands to try to work them apart as you drop them into a couple of Tablespoons of hot oil. Spread them around to make sure they all get coated with oil initially, so they don't stick as they start to cook. Stir them occasionally without covering the pan to distribute heat and moisture. It's important that the moisture evaporates evenly.
As the onions start to form a thick brown liquid, I will sometimes augment the natural 'caramel' in the onions with some actual caramel, by sprinkling in a Tablespoon or so of white sugar. Brown sugar would be delicious too.
Caramelized onions can form the basis of a sauce, a stew, or can make a nice garnish for meats or other dishes.