I'd be upset with Kawasaki not Deere.
My troubleshooting ways are unorthodox probably because of 43 years of that I had to be creative.
It's either fuel or spark problems. I'd start with fuel.
On ebay you can order these universal 12V fuel pumps for $10. I'd get one and several feet of 5/16" fuel hose. Add a few feet of wire to the two pump leads with clips on end.
Get a 1 or 2 gallon gas can (new or 100% clean). Zip tie or mount (away from heat

), fuel hose in...to pump in...hose out run directly to carburetor. Plug & clamp removed hose line.
Connect pump leads directly to battery (red+, black--).
See how it runs now.
If it still cuts off you've eliminated fuel, so reconnect original fuel line.
Now spark. Two cylinder engine has two magneto coils each with a 1/4" kill tab. Unplug both tab leads.
Get two 1/4" female wire connectors and an SPST switch. Make a "Y"...connector each end to plug in kill tabs...then to switch...other switch connector a wire to ground. Switch open (off) you have spark. On kills engine.
See how it runs now.
If it runs fine with subbed-in fuel system then troubleshoot what's on mower (hoses, vapor-lock, etc). If it runs fine subbed-in ignition troubleshoot that...rubbed wire insulation, safety switches, ignition switch.
If it still acts up I'd be thinking carburetor. Maybe carb should have a heat shield.
If it runs fine for a few hours then starts sputtering & dying I'd have starting fluid (od safer...WD-40) with me and spray around carb checking for air vacuum leak.