PC-12 pilot reported engine failure in cruise flight. Pilot improperly restarted engine which subsequently malfunctioned; but pilot landed safely.

Narrative

This is what I remember of the engine failure on the PC-12 on this day.After takeoff a CAS message appeared; Autothrottle (AT) Fail. After leveling off at FL300; I went through the QRH to look at the checklist for this failure. I then did a search of the POH for AT Fail and landed on a page which contained a table of all the cyan CAS messages. I found AT Fail in this table and looked at the top vertical listed items. I thought maybe I did something wrong causing the AT failure and used the vertical items of this table to check the airplane. When I saw in this table; Engine Start; I then looked at the overhead panel and saw the engine run switch was up. I instantly thought I made a mistake leaving this on and turned it off. After a second; the engine shut off. I then put the engine run switch back to run; waited for a second or two; checked the engine gauges and saw the engine continued to spool down. By this time the propeller was feathered and I pushed the Start button. Nothing happened so after another couple seconds I pushed the Start button a second time. This time the engine started. It spooled up and the propeller came out of feather. As the airplane started to accelerate and climb; a loud bang was heard and out the corner of my left eye; I saw small parts of the engine go by. I [requested priority handling]; and requested information on a close airport that's VFR with a long runway. The controller stated there was one about 50 miles away. I was given a heading of about 100 degrees and turned to it. After this I requested the passenger to come up front. I had the passenger sit in the right seat and instructed her to fly the heading bug and the Dynamic Speed Bug (green airspeed carrot). I programed the FMS for Direct to the airport; selected a visual approach with a 4.0-degree GP and 3.5 mile final to the runway. I then referred to the QRH for an Engine Failure in Flight. I took the controls back from the passenger and flew the airplane to the airport with a landing at about the 1;000-foot markings and rolled out to the right on a high-speed taxiway where emergency crew were parked. I shut down all the electrics and we exited the airplane.I noticed some paint chips missing from the top of the left wing and no other damage.After the plane was in the hangar; I inspected the engine exhaust and saw metal pieces laying in them. I also saw some paint chips on the right-wing fairing to fuselage. The left wing deice boot also had several cuts and divots. There was no fire; oil leaks; or other visible damage as viewed with the engine cowling open.On return to home; I reviewed the POH and QRH. It's obvious I screwed up;1. I should have left the AT Fail alone as the QRH stated. 2. I should not have shut the engine down and thought before a quick knee-jerk reaction.3. Once the engine was shut down; I should have referred to the QRH and followed the Air Start checklist.I feel the error of turning the engine off was partially caused by flying airplanes for decades and thousands of hours that had a lever or physical control to shut the engine down. I just completed Aircraft Y training which may have further wired my brain to not immediately associate a switch to shutting the engine down. I've read many NTSB reports over the years and when I read a pilot did something stupid; I'd usually think; how could he do that? Don't think for a second you are not capable of doing something stupid. Never do a keen-jerk reaction if you have altitude and the plane is flying fine. Think about the switch or control; what does it do; what does it control before changing it (outside of flow items). Follow checklists and don't stray from them thinking you are smarter than it. Also; take your time before moving a switch or control; even when following a checklist and especially if not following a checklist (outside of flow items).I feel the Engine Switch should be harder to shut off; require a different motion (rotary) than all the switches onthe overhead; better distinguished with red and best; to have it near or behind the power lever to better associate it with the power level. This is my true statement to this incident.