Prep the liver: Pat the slices dry with paper towels.
If you prefer a milder taste, soak in milk or buttermilk for 30 minutes in the fridge, then drain and dry well. Trim any visible membranes.
Season and dredge: Season both sides with salt and pepper. Place flour on a plate and lightly coat each piece, shaking off excess.
A light dredge helps browning and a silky pan sauce.
Caramelize the onions: Heat 1 tablespoon butter and 1 tablespoon olive oil in a large skillet over medium to medium-low heat. Add the onions and a pinch of salt. Cook, stirring occasionally, until golden and soft, about 15–20 minutes.
If they brown too fast, lower the heat.
Build the onion sauce: Add 1/2 cup broth to the onions and let it reduce by about half. Stir in Worcestershire and Dijon if using. Transfer onions to a bowl and keep warm.
Sear the liver: In the same skillet, add 1 tablespoon butter and 1 tablespoon olive oil over medium-high heat.
When hot and shimmering, add liver in a single layer. Do not crowd; cook in batches if needed.
Cook quickly: Sear 1–2 minutes per side for medium-rare to medium, depending on thickness. The center should be slightly pink; overcooking makes liver tough. Remove to a warm plate and tent loosely with foil.
Deglaze and finish: Lower heat to medium.
Add remaining 1/2 cup broth to the pan, scraping up browned bits. Whisk in the remaining 1 tablespoon butter. Return onions to the skillet and simmer 1–2 minutes until glossy.
Brighten: Off heat, stir in vinegar or lemon juice to taste.
This balances richness and cuts any bitterness.
Combine and serve: Nestle the liver back into the onions for 30 seconds to warm through. Taste and adjust salt and pepper. Garnish with parsley and serve right away.