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

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

Cook onions uncovered, until just al dente, about 10 minutes. If liquid dissipates too quickly, add more water, a 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 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.