Heat 4 tablespoons of butter in a large nonstick skillet over medium-high heat. When the butter has finished foaming, add the onions, along with a generous pinch of salt. Stir so that all of the onions are coated in butter. When the onions begin to release their liquid, reduce the heat to medium-low and cook, stirring frequently, until tender and a light golden brown. You are not looking for deep caramelization here, but you do want a definitive color change. This may take 10-15 minutes. When the onions are done, remove them to a bowl and set aside.
Season the liver slices on both sides with salt and pepper and dredge in the flour, shaking off the excess; you just want a light coating. Melt the remaining butter in the same skillet, no need to have cleaned it out, and when the butter stops foaming, add the liver slices in one layer. You may need to do this in batches. Cook about 1-2 minutes per side until the liver is golden brown but still soft. Once you have turned the liver twice, ensuring that there is good color, remove the liver to a warmed plate, and if you need to, sauté the second batch of slices. Try to not over-cook the liver as it will become tough.
Once all of the liver is browned and set aside, return onions to the pan and add stock, and lemon juice. Cook, stirring, until the juices have thickened slightly, then taste for seasoning, add salt, pepper and lemon to taste. When the sauce tastes good to you, carefully lay all of the liver over the onion sauce mixture and gently spoon some of the sauce and onions over the top of the liver, cooking about 4-5 minutes longer to finish cooking the interior of the liver. You can check for doneness by making a small cut into the thickest part of the liver, you are looking for a gentle pink with brown on the edges, and no raw or rare looking line in the middle.
Once the liver is cooked, serve with a generous tangle of onions on top, with the sides of your choice—mashed potatoes or buttered rice highly recommended for soaking up the sauce.