Drawer/Cabinet Pull Install Splintered interior Veneer
I agree that it sounds like crappy workmanship that came from your contractor. The best situation is to talk to the contractor and tell them that how unhappy you are with the crap quality.

What the contractor should have done when drilling the holes for the drawer pulls was to clamp a piece of scrap wood to the inside face of the drawer front to prevent the splintering as the drill bit exited through the wood. Not doing this is the sign of a rank amateur contractor. 

If this has just happened on the drawers and not on any doors there is one thing that may bring some piece of mind for you. Rarely are people going to be looking down inside the fronts of your cabinet drawers so maybe you can just let it go after getting some financial adjustment out of the contractor. 

Note that most cabinet drawers are not built in a way where it is particularly easy to remove and replace the drawer fronts. Particularly in the case where higher quality drawer assemblies may have glued joints. Some cheaper cabinet work may just have the drawer front stapled in place through the side panels. This latter may be easier to replace. 

If the damage exists on cabinet doors then you are probably facing having to have the whole door replaced if you cannot accept the crappy quality work.