Beechjet 400 Captain reported an in flight shut down of the right engine during cruise when the FADEC buttons were pressed to select A Channel. The Flight Crew descended and were able to restart the right engine and continue to destination airport.

Narrative

While in cruise flight between ZZZ1 and ZZZ2 I was discussing the Nextant Williams engine systems with the Second in Command (SIC) (which are different than the original P&W JT-15D equipped Beech jets). The engines were on the 'Channel B' Full Authority Digital Engine Control (FADEC). I stated that either channel could be selected at any time. At that time I pressed both FADEC channel buttons to switch from the 'B' channel to the 'A' channel. The left engine continued to operate normally; however the right engine shut itself down. As a result we requested a descent from ATC to maintain speed while locating the appropriate checklist. We continued to descend into the starter-assisted air start envelope; leveling off at 28000 ft. An attempt to restart on the channel A FADEC was unsuccessful. The channel B FADEC was selected on the right engine and the restart was successful. Once the restart checklist was complete we climbed the aircraft back to 40000 ft. and continued the flight to ZZZ2 as planned.We did not report the engine issue to ATC; so as not to cause undue concern. We would have reported it had a restart not been successful. We experienced some difficulty in accessing the appropriate checklist as there is not an FAA approved cockpit checklist for a Nextant (other than the original Beech jet 400A checklist; which does not have the proper/revised procedures for the Nextant). The electronic Nextant checklist installed on the PL-21 avionics only had normal procedures and no abnormal or emergency procedures. The original AFM and POM are also unmodified. There is a supplemental Nextant manual which has the appropriate changes and checklists; however the print is very small and not tabulated like the original Beech jet cockpit checklist. Since CAE trains pilots using their (for training purposes only) cockpit checklist; this was the first time I had to access the supplemental manual in flight; which caused me to realize its difficulty for cockpit use (especially at night).