I've flown several times with a broken hand. I've done it in crewed and single pilot cockpits, including stick and rudder type cockpits while flying ag (crop dusting). I think it's not really possible to say you can or can't with a broken hand, because there are just too many variables.
I broke my hand a few years ago while doing some Learjet work. I rode a motorcycle to a clinic where it was x-rayed, and I was told it wasn't broken. It was on new year, so I rode my bike to a local drop zone, went skydiving, did some work, took a flight, and when I got home, I had a message asking me to call the clinic. I was told that they had changed their mind, and yes, my hand was broken. It hurt; it was uncomfortable, but didn't interfere with my ability to fly, deploy a pilot chute, ride a motorcycle, etc. A different kind of break might have been handled quite differently.
I did go to the trouble of securing a statement from the physician stating that the injury posed no danger.
On another occasion a bolt broke while I was working on an airplane. The wrench I was moving was on the bolt, and when the wrench slipped, my finger was sliced open to the bone and my finger was broken. We were on a very tight schedule, and no immediate medical facilities presented themselves. I put antibiotic in the wound, closed it with butterfly sutures, broke a tongue depressor in half and put it between the injured finger and the adjacent one, and taped them together with bandage tape. Then went back to work. I was able to turn wrenches with a fair amount of discomfort but no loss of ability, and to fly.
When I was able to reach medical facilities later, I was unable to bend that finger properly, and it stayed that way for about a year. It didn't interfere with my flying, and the AME had no problem with it. It just hurt.
