Dredge the liver and avocado slices with the flour seasoned with the salt and pepper. Heat half of the butter and sauté the liver and avocados very quickly on both sides. Transfer to a warm platter and keep hot.
Heat the remaining butter in a saucepan until lightly browned. Add the lemon juice, stock and thyme. When hot, pour over the liver and avocados. Sprinkle with parsley.