Rinse the beef liver under cold water and pat dry with paper towels.

In a shallow dish, combine the flour, salt, pepper, and paprika. Dredge each slice of liver in the flour mixture, shaking off any excess.

Heat 2 tablespoons of olive oil or butter in a large skillet over medium heat. Add the sliced onions and sauté for about 10 minutes, stirring occasionally, until they are soft and caramelized. Remove the onions from the skillet and set aside.

In the same skillet, add the remaining tablespoon of olive oil or butter. Increase the heat to medium-high. Once hot, add the liver slices to the skillet in a single layer. Cook for about 2-3 minutes on each side, or until browned and cooked through.

Return the caramelized onions to the skillet with the liver. Drizzle with balsamic vinegar if using, and stir gently to combine. Cook for an additional minute to heat through.

Remove from heat and garnish with fresh parsley before serving.