Set a heavy pan to heat over very low flame.

Chop bacon very fine. Place in the pan, and let cook very slowly, to render out the fat. (Depending on the pan and the stove, this will take at least 10 minutes, and as long as 20.) Or - heat oil in pan.

Peel and quarter onions. Slice thinly with sharp knife or mandoline.

When bacon is fully rendered, add onions to the pan, and stir well, to cover them with fat. Then let it cook down over low heat, stirring every 10 minutes or so, until onions are caramelized and cooked down. This may take an hour or longer.

When onions are cooked down, add balsamic vinegar. Stir it in well, then continue to stir for a few minutes to cook onions down even further, and work in the vinegar.

When done, remove onions, and deglaze the pan with 1 T water. Add to the onion mixture.