That's probably going to be either fuel supply or ignition.
This is what I would do - go to autozone, o'rielly's or some auto parts store that has a loan-a-tool program. Borrow their fuel pressure gauge. Pull the passenger seat and fuel tank plastic cover. Install the fuel gauge in the line from the tank. Start the machine. The gauge should indicate 50 psi +/- a couple psi. Then run the machine til it is hot so that it is sputtering as you describe and check the fuel pressure. It should be the same. If the fuel pressure stays the same then your pump and the sock filter on the bottom of the pump are ok and fuel delivery is good, unless there is a problem with your injectors. The injectors are probably ok. If the fuel pump is bad, there's an orielly part number floating around which is a more affordable replacement.
In general, the problem you describe sounds a lot like an ignition coil going bad, which is exacerbated with heat. I've had problems on other small engines where the coil was bad and the machine would run fine until hot and then if you turned it off, it would not restart until it cooled down, or, as you describe it ran rough and died. So, that may be an area to look. I don't know how much coils are from the dealer but there are aftermarket coils on ebay and amazon for around $25. I would replace the plug wires too if replacing the coil.
Spark plugs are something cheap to switch out, although if it's starting, they're less likely to be the problem.
I would also pull the exhaust tip and then pull out and clean the exhaust screen.
Someone else may have some other thoughts.