In a large skillet, heat the oil and half the butter. Add the onions.
Cover and cook over low heat for about 30 minutes or until the onions become translucent and soft, without getting browned. Season with salt and pepper.
In another skillet, brown the veal liver, one minute per side, a few slices at a time, in the remaining butter. Season with salt and pepper. Set aside on a plate.
Add the cooked liver with the hot onions and cook for 1 to 2 minutes or until the liver is medium-rare. Remove from the heat, add the parsley and lemon juice. Adjust the seasoning and serve with Mashed Celery Root (see recipe). It is traditionally served with polenta.