1. In a large nonstick frying pan, heat 3 tablespoons of the oil with 1 1/2 tablespoons of the butter over moderately high heat. Add the onions and cook, stirring occasionally, for 10 minutes. Stir in 1/4 teaspoon of the salt. Reduce the heat to moderate and cook, stirring frequently, until the onions are well browned, about 5 minutes longer. Remove.
2. Sprinkle the liver with the remaining 1/2 teaspoon salt and the pepper. In the same pan, heat the remaining tablespoon oil and 1/2 tablespoon butter over high heat. When the pan is very hot, add the liver and sage. Cook, stirring, until just done, 1 to 2 minutes. Remove from the heat, return the onions to the pan, and toss.
How to Cook Liver: Forget all the bitter, grainy liver you may have had. The way to make liver taste good is primarily a matter of resisting the temptation to overcook it. It should be seared to keep the juices from escaping and should be slightly pink inside, not dingy gray. Use high heat bravely and briefly.
Wine Recommendation: Brunello di Montalcino is Tuscany's most exalted wine because it offers a combination of richness, power, and elegance unmatched by any other in the region. Serve one here for a sumptuous experience. A more affordable option is Rosso di Montalcino, a sort of scaled-down brunello.