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 and browned bits in the pan.

Add the olive oil to the same skillet and increase the heat to medium-high. When the oil is hot and shimmering, add the floured liver in a single layer (work in batches if needed). Fry until browned on the outside and still slightly pink in the center, about 2-3 minutes per side. Remove the liver and set aside.

Reduce heat to medium. Pour in the beef broth. Use a wooden spoon to scrape up all the browned bits from the bottom of the pan.

Return the onions and liver to the skillet. Stir gently to coat everything in the sauce. Reduce heat to medium-low, cover, and cook for 15-20 minutes. If using calf liver, 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.