So, first of all, this solution will be cleaner and easier if you could power the first Shelly all the time and wire the switch into the switch input. But if that's not possible, you can still make it work.
If you can wire the Shelly for permanent power, I think you can even make it work with the original Shelly firmware. It can call webhooks so the first Shelly can call the HTTP API of the second Shelly. I didn't try it out though.
Otherwise you'll have to flash Tasmota: Original Shelly series Shelly Plus series
Then you can configure the first Shelly to periodically (e.g. every 55 seconds) call Tasmota's HTTP API on the second Shelly and use PulseTime
on the second Shelly to turn off after 1 minute.