Creating Caramelized Onion Beef Liver is straightforward if you follow these steps closely:

1. Prepare the Onions: Heat 2 tablespoons of olive oil and 1 tablespoon of butter in a large skillet over medium heat. Add the sliced onions and sprinkle with a little salt.
2. Caramelize the Onions: Cook the onions, stirring occasionally, until they become soft and golden brown. This should take about 10-15 minutes. If they start to stick to the pan, reduce the heat.
3. Remove Onions: Once caramelized, remove the onions from the skillet and set them aside on a plate.
4. Cook the Liver: In the same skillet, add the remaining butter. Season the beef liver slices generously with salt and pepper.
5. Sear the Liver: Raise the heat to medium-high. Once the skillet is hot, add the liver slices. Cook for about 2-3 minutes on each side, or until browned and slightly firm but still pink in the center.
6. Add Balsamic Vinegar: If using, sprinkle the balsamic vinegar over the liver while in the skillet for added flavor and glaze.
7. Combine: After cooking the liver, return the caramelized onions to the pan and toss them together briefly. Allow them to warm through.
8. Adjust Seasoning: Taste and adjust the seasoning as necessary, adding more salt and pepper if desired.
9. Plate the Dish: Carefully plate the liver and top with the caramelized onions. Garnish with fresh parsley for color and extra flavor.

With these clear instructions, you’ll be able to create a dish that not only looks stunning but also bursts with flavors.