Rinse the liver under cold water and pat dry with paper towels. If slices are large, cut into more manageable pieces. In a shallow dish, combine the flour, salt, and black pepper. Dredge the liver in the seasoned flour, coating both sides and shaking off any excess. Slice the onion into thin rings or half-moons.

In a large skillet over medium heat, melt the butter. Add the sliced onion and cook, stirring occasionally, until softened and lightly golden, about 8-10 minutes. Transfer onions to a plate, leaving the butter in the pan.

Add the olive oil to the same skillet, keeping the heat at medium. When the oil is hot and shimmering, add the floured liver in a single layer (work in batches if needed). Fry until browned on both sides, about 2-3 minutes per side. The center will still be pink, and that is fine, it finishes cooking in the sauce. Leave the liver in the pan, returning any earlier batches to it now.

Pour the beef broth in around the liver and use a wooden spoon to loosen anything stuck to the bottom of the pan.

Return the onions to the skillet and spoon them over the liver. Reduce heat to medium-low, cover, and cook for 15-20 minutes. If using calf liver, or liver that was previously frozen, begin checking for doneness at 10-12 minutes. The liver should be tender and holding together, not falling apart.

Stir in the apple cider vinegar. Serve hot with the pan sauce spooned over the top. Great with mashed potatoes, rice, or your favorite side.