Paul Verdun is a restaurant mogul in Paris whom Hassan befriends. And while he serves as a mentor for our main man—here we can think of him as filling Mallory's shoes once Hassan sets off on his own—perhaps most importantly, he stands at the center of a huge debate about tradition versus change that comes to the fore repeatedly in the novel.
Paul has spent his whole life becoming a big shot among the legends of traditional French cuisine. This is the same tradition that bred Mallory, and it's also the same tradition that Hassan has studied and been inspired by. So when you think Paul, feel free to consider him the president of Team Tradition—he's ahead of Hassan in his career, and he's more esteemed than Mallory.
As Paul gets older he struggles to keep up with the changes. The new techniques don't build upon the past, and instead they totally change the game. As the world of haute cuisine changes things up, the old school ways become obsolete—and as they do, so does Paul. He's the one still painting real figures after Picasso comes along and starts painting everything in squares and squiggly lines.
Here's the sad part: Eventually Paul can't keep up anymore and takes his own life. He becomes a lesson to Hassan, who is stuck facing many of the same battles. The moral of Paul's story as Hassan understands it, then, is that you have to stay above the tide or you'll get pulled under.
Despite his clear fondness for Hassan, Paul's still a bit of a mystery to the younger chef. As Hassan explains:
I don't want to exaggerate our closeness; I don't suspect anyone, not even his own wife, ever cracked the high-octane energy of that man. (13.83)
In other words, Paul's the kind of guy that nobody gets super close to. He's sort of a Jay Gatsby figure, who swoops Hassan up in his private jet and takes him to dine on fine food and champagne at the drop of a hat.
Despite his mysterious ways, Paul finds real friendship from Hassan. He believes they are both "'made from the same ingredients'" (15.45), and after his death—despite Hassan's understanding of Paul as a pretty opaque dude—he still goes back and forth between remembering Paul his friend and the famous chef, Paul Verdun.
Paul's troubled and intensely passionate nature is revealed on a couple of occasions. One such occasion is when he and Hassan both visit the Musée D'Orsay, and Hassan finds Paul lost in front of a huge painting called The Excommunication of Robert the Pious. After a couple of minutes, Paul comes out of his reverie, but he never explains why he was staring at the painting. In hindsight we get it: Paul probably sees himself as the king being excommunicated, though he keeps this feeling of failure to himself.
After Paul dies, his true fondness for Hassan is revealed, along with his intensely good heart. In his will, he instructs Hassan to organize a huge memorial dinner and invite everyone who's anyone in the culinary world. The real reason isn't to throw a heckofa party to honor the celebrity chef, though—the real reason is that in doing so, Hassan will become known and admired by a whole lot of powerful culinary folks. In other words, Paul uses his death to secure Hassan's career. Pretty sweet, right?
So let's recap. Paul provides a cautionary tale to Hassan. He is the person that has the money and success and fame, but doesn't have the strength to control it and so it destroys him instead, as both a person and an artist. As Hassan gets older, he admits that he starts to go crazy with stress and worry and feels like "[Paul's] spiritual malaise had jumped bodies and entered me, like some flesh-eating parasite from a Hollywood horror film" (16.7). Kind of like a weird martyr for the cause, Paul's paves the way for Hassan's success on both a personal and professional level.