"Intellectual directors", "mind game". These phrases can have multiple meanings.
For instance, you can call Robert Rodat (and Steven Spielberg) a genius for 'Saving Private Ryan'; in particular that scene in the movie of Omaha beach. The emotions, horror, and everything about that scene makes it one of the greatest of all time.
Christopher Nolan, too, can be called intellectual, but in a very different way, for his masterpiece Inception.
Similarly, Christopher Markus and Stephen McFeely, along with Russo Brothers, are other names that you can call "intellectual" for their once-in-a-lifetime Avengers: Endgame extravaganza. The movie, although based on fantasy, with so many storylines intricated so closely, it sure looks very clever on many turns.
So, there are different meanings of "intellectual" for the directors and Screenwriters. Some can turn exuberating action and fantasy into something rational, others use emotions and somber setups to create a masterpiece.
Besides, a movie's class comes not solely out of a director or writer BUT the entire crew and cast.
For example, Woody Allen is considered one of the greatest directors of this generation. However, his "Shadows and Fog", considered one his worst, doesn’t come close to that recognition.
So, when a movie is successful, interesting, intelligent and completely outstanding, it's not just about the director or screenwriter. It's about the collective work of the entire cast and crew.
In that context, here are some of the intellectual movies (or, at least, the clever ones):
(Courtesy: Narrative First)
- Goodwill Hunting
- The Martian
- The Imitation Game
- Fight Club
- The Theory of Everything
- Minority Report
- A Beautiful Mind
I am sure there are many more intellectual movies out there. These are the ones I can come up with. Besides, since the word "intellectual" itself is quite subjective, different people can find different movies intellectually stimulating. Some would like the Space fantasy more, others can find Drama much more connecting.
I hope this answers your question.