Pretty simple premise. Both have directed almost exactly the same amount of feature films. Both are staples of 00s/10s in their respective genres. Chronologically film by film. Which film do you prefer.
Alien³ vs Following (wins)
Se7en (wins) vs Memento
The Game (wins) vs Insomnia
Fight Club vs Batman Begins (wins)
Panic Room vs The Prestige (wins)
Zodiac vs The Dark Knight (wins)
The Curious Case.. vs Inception (wins)
The Social Network (wins) vs The Dark Knight Rises
The Girl with the Dragon Tattoo vs Interstellar (wins)
Gone Girl (wins) vs Dunkirk
Mank vs Tenet draw
6-4-1, Nolan wins for me. Nolan was better in the 00s, and Fincher was better in the 10s.
Feel free to copy and paste and then just put the film you prefer in bold.