Preheat the broiler. In a 12 to 14-inch saute pan, heat the olive oil over medium-high heat until hot but not smoking. Add the sliced onions and cook until lightly golden, about 3 minutes. Season the liver pieces with salt and pepper and dredge lightly in the flour. Add the liver slices and let cook until lightly browned on 1 side, about 3 minutes. Turn the liver over and cook the other side until equally brown, about 2 minutes. Deglaze with the red wine vinegar. Salt and pepper, to taste. Remove from the heat. Pour soft polenta on center of the plate and spoon the liver and onions on top.
In a 3-quart saucepan, heat water and salt until boiling. Drizzle in polenta in a thin stream, whisking constantly, until all polenta is incorporated and mixture begins to thicken. Switch to a wooden spoon, pull off heat and continue to stir until thick as paste.