Soak the Liver: Place the sliced beef liver in a bowl of milk. Let it soak for 30 minutes to remove bitterness.
Prep the Ingredients: Pat the liver slices dry with a paper towel. Season lightly with salt and pepper, then coat with flour on both sides.
Cook the Onions: Heat 1 tablespoon of butter and 1 tablespoon of olive oil in a large skillet over medium heat. Add the onions and cook, stirring occasionally, until caramelized (about 10 minutes). Remove the onions and set aside.
Cook the Liver: Add the remaining butter and olive oil to the skillet. Increase the heat to medium-high. Add the liver slices and fry for 2-3 minutes per side until browned and cooked through.
Combine and Serve: Return the onions to the skillet. Stir gently to mix and heat everything through. Top with fresh parsley or sliced green onions, if preferred.
Serve: Plate the liver and onions hot with your favorite sides like mashed potatoes or a fresh salad.