Place the liver slices in a shallow bowl and cover with milk. Let it soak in the refrigerator for 30 minutes to 1 hour to reduce bitterness and tenderize the liver.

In a large skillet, melt 2 tablespoons of butter over medium heat. Add the sliced onions and cook for 15-20 minutes, stirring occasionally, until they are soft and caramelized. Remove the onions from the skillet and set aside.

In a shallow dish, combine 1/2 cup flour, 1/2 teaspoon salt, and 1/4 teaspoon black pepper. Mix well.

Remove the liver from the milk and pat it dry with paper towels. Coat each slice in the seasoned flour mixture, covering both sides.

Heat 2 tablespoons olive oil in the same skillet over medium-high heat. Cook the liver in uncrowded batches for about 2 to 3 minutes per side, adjusting for thickness, until the thickest piece reaches 160°F (71°C) on a food thermometer.

Return the caramelized onions to the skillet, spooning them over the liver slices. Allow the dish to heat through for about 1 minute.

Transfer the liver and onions to a serving platter. Garnish with chopped fresh parsley, if desired. Serve hot with sides like mashed potatoes, rice, or a green salad.