Place sliced beef liver in a bowl and cover with milk. Let rest for 15–20 minutes in the refrigerator to mellow out any strong flavors.

In a shallow dish, whisk together flour, salt, and pepper. Drain liver, pat dry, then coat each slice evenly.

Heat 2 tablespoons of oil and 2 tablespoons of butter in a large skillet over medium heat. Add sliced onions and cook, stirring occasionally, until soft and golden (about 10 minutes). Transfer onions to a plate.

Add remaining oil and butter to the skillet. Increase heat to medium-high. Working in batches, cook liver slices 2–3 minutes per side until browned but still slightly pink inside.

Return onions to the pan, toss gently with the liver for 1 minute. Remove from heat and sprinkle with chopped parsley.