IMDb uses an algorithm to determine the Top 4 movies a director is known for. I'm not sure how much you agree, but it's all up to IMDb, and that is the source I used.
In this case, I've decided to only include films that these directors actually directed. They may have acted in the film they directed, or other films, written the film, and/or produced the film, but the films among these lists are only films that the featured Oscar-winning director directed. i.e. If Mel Gibson were an option, and the hinted year was 2004, the answer would be The Passion of the Christ, which he directed. It wouldn't be Paparazzi, which he produced, but did not direct.