Peel the onions, leaving and washing any root strands you may find.

In a 12 to 14-inch saute pan over a medium high flame, heat the olive oil until just smoking. Add the butter and cook until the foam subsides. Add the onions and saute until light golden brown on all sides, about 8 to 10 minutes. Add the sugar, vinegar, rosemary, and water and bring to a boil.

Cook the onions uncovered, until just "al dente", about 10 minutes. If the liquid dissipates too quickly, add more water, 1/4-cup at a time, realizing that it is essential not to overcook the onions. The sauce should just adhere to the onions. Remove from the saute pan to an earthenware dish and hold in a warm place, or allow to cool if you are serving them later or as an antipasto