I agree with Euclidean. Where there is a will, there is a way. There are even neck holsters for the P3AT.
That said, I'm not a pro chef but I'm the cook in the home. And since I carry 24/7 (where legal gandog56) I can tell you some of my don'ts.
Center line carry = BAD. Hot pasta water on your gun flushes out a lot of the good lube needed to keep the smokewagon happy. Although it did divert the water and keep it from getting on other things. So you may want to think about Center Line anyway.
Carry a gun with a good finish. It's gonna get dinged, splashed, drenched, and dropped. And I'm not even moving at line cook speeds. Tomato sauce does a number on some metals. Kosher salt is NOT a gun finish’s friend. Oh, and blood is not a friend to bluing. But you are a pro so you've never cut your fingers.
My Hard Chrome P3AT is usually my Cooking gun (not to be confused with my Bar-b-que gun). It's small, cheaper than any of my others, and pretty tough. So when I screw it up (and as clumsy as I am, I will), it won't kill me to replace.
But I have to agree with others. You are in a professional kitchen with a lot of other hot angry line cooks. There are literally dozzens of better close quarters combat options hanging from the hooks in there. I pity anyone who bursts in on a bunch of cooks in a professional kitchen. He might wind up as the lunch special the next day without a single shot fired.