Cut a check, invoice, work order, military order.  It's from days before electronic reproduction.  Folks used to "cut a record" to carve a physical groove, and then  they "cut a tape" which was a grooveless carryover.  (But still sometimes groovy.)  Today we "burn" CDs.
As for checks and orders and such, old mimeographs were hand-cranked and used cut stencils.