Cut liver lengthwise into 4 long pieces, then, using a very sharp knife and pressing the palm of your hand firmly against the meat, slice each piece crosswise into pieces as thin as possible.
Heat 4 tbsp. of the olive oil in a large skillet over medium heat. Add onions and cook, stirring frequently, until soft and deep golden brown, about 20 minutes. Transfer onions with a slotted spoon to a bowl and set aside.
Increase heat to medium-high and add remaining 2 tbsp. oil. When oil is sizzling hot, add liver and cook, in batches to avoid overcrowding the skillet, stirring constantly with a wooden spoon, until brown and crispy on the edges, 3-5 minutes. Season liberally with salt and pepper, then add reserved onions and accumulated juices. Cook for 2 minutes, stirring and turning liver and onions constantly while shaking skillet over heat. Transfer to a heated serving platter.
Add butter to skillet and scrape up any brown bits stuck to bottom of skillet as butter melts. Remove skillet from heat and stir in parsley. Spoon butter and parsley over liver and onions. Serve with Grilled Polenta, if you like.