You can with careful preparation but you still stand a chance of it exploding in the planer. This may damage a new planer which you would not like that to happen. Your best options are a router jig to plane the cutting boards until you can get a drum sander.
It is easier than you think. You glue strips of scrap wood, cut that into more strips and rotate every other strip 90 degrees, glue these strips together, cut the strips and repeat gluing process. The last step is cut the glue up across the grain and turn up on end alternate the strips and glue back together. Then sand or router flat. Thanks for the compliment.