Surfacing a end-grain cutting board on a router table, or with a router is fairly easy and quick. Using a hand held router is the same as surfacing any other board so I won't go into that, there is plenty of good information on that technique all ready. One note if you have a small router table you may need to clamp some boards to your table for the rails to ride on.
First you will need two rails to attach to the sides of the cutting board. I used 1x4s little longer than the cutting board, and a couple of gage blocks normally I use 1/4 or 1/2 inch just nothing to thick unless you have a tall bit.
I set the cutting board on the gage blocks and place the rail along the sides on their ends, then I pin nail them to the board, and then I like to put a couple of #6 screws in the center of each rails to be sure they don't move. it is important that the rails don't rock. because they are your referenced surface.
I use a 3/4 Straight bit, but a 1/2" mortise bit gives a better finish or one of them flat bottom pattern bit that are designed for routing bowls would do a great job I am betting.
Then all you do is set the bit a little higher that the gage block you used to set your rails, I started about 1/32nd and then slowly, with your hands on the top part of the rails, slide the cutting board back and forth across your table letting the bit do the work.
Don't try to remove to much at one time because the bit can grab and the board can get away from you, it is better to take light cut and be safe. Stop and check your progress, you will more than like miss some sections just keep going over it. Just a note the first few times you get in to the rail it will make a lot of noise and grab the piece to be mindful of that, and it is a good idea to mark on your table a line where your cutter is you that way you will know where you are cutting.
The end product is fairly smooth and flat, but it still needs a little attention if you missed any little spots they come off quickly with a sharp chisel and then you can clean up the lines with a sander or a card scraper to get it smooth
This bring to another point I used my small router table that I keep set up with a 1/4 round over bit. I did this to show that even with a small table you can still use this technique I know a lot of woodworkers own these type of router table and I didn't want them to think they needed a big fancy table to do it. all I did was clamp a couple of straight boards to the front and back and I used a spring clamps to hold them while I lined them up with my table then clamped them down with a couple of bar clamps. This gave the rails something to ride on while the bit surfaced the board.