You can build a frame around the cutting board using spacers between the cutting board and the frame to locate the groove the desired distance in from the edge. The spacers need to sit at the same height as the cutting board to keep the router from tipping. Anchor the frame to a piece of ply or mdf and make sure there is no play between the board, spacers, and frame. Feed the router from left to right (clockwise) and don't try to feed too fast. Pick a round nose or core box bit of the appropriate diameter, they are some of the cheapest bits normally.
Someone I consider a master woodworker once told me that a master woodworker is not someone who never makes mistakes. He is someone who is able to cover them up so that no one can tell.