How can I make carving a chicken less messy?
As suggested in comments, you could try putting a small cutting/carving board into a large pan or baking tray to hold the liquid better.

In general, this is a good problem to have.  If you are carving a whole chicken, and no juices are coming out at all, you've probably overcooked it.  Resting will only help so much, as fully cooked chicken will generally be at a temperature when muscle fibers will have tightened and contracted, thereby squeezing out some moisture.  It's also possible that your chickens may have been injected with additional brine/broth (though this will generally be indicated on packaging).

As for how to prevent it, I might suggest less carving on the board and more cutting on individual plates.  I'm not sure how you cut up a roast chicken, but I generally start by only cutting off about six pieces -- the wings, the leg/thigh quarters at the thigh joint, and with a little practice you should be able to remove about 80%+ of the breast meat intact from each side in one large filet.  Depending on the size of your chicken, how many people you are planning to serve and what preferences they have, you can then go further and separate the legs from the thighs, and/or do a few cuts through the entire breast to make more white meat portions.  The fewer cuts you make on the carving board, the less juices will leak out there.  Instead, you can give more whole cuts (or parts of whole cuts) to people, who can cut them apart on their plates.  By that point, they meat will have rested and cooled more so it will release less juice, and any juice that is released will be flavor on their plates too, perhaps to be sopped up by other means.

Traditionally, I know many people tend to think of "carving a chicken" the same as "carving a turkey" and slicing thinly through the breast.  To me, that's a lot of work and often results in cold chicken by the time it is served.  It also will tend to release more juices during the slicing.  Even with a 20 lb. turkey, I tend to cut the entire breast off whole, then make thick slices down through the breast after it is detached and on the board.  It keeps the meat warmer, allows everyone to have a bit of skin if they want, and keeps everything moist.  Unless you or your guests are really attached to very thin slices of meat, I find it's better to leave things as whole as you can when you put them on people's plates.