Ive seen cutting boards with metal handles installed from the side faces that don't appear to have any outside facing screw holes.. how the heck are they doing this?
Based on the board linked to in the Comments above I'd say that style of handle is almost certainly glued in place.
You could pin then, with a piece of steel or brass rod (similar to how some knife handles are held in place), but with the right adhesive that sort of mechanical fixing is not at all necessary on anything that doesn't have to take much strain.
Best glues to use here I think would be epoxy (especially useful if the fit of the metal in the holes is a little sloppy) and polyurethane adhesive, like the original Gorilla Glue. Note that despite polyurethane adhesive foaming up and appearing to fill gaps it is not a gap-filling glue, and is only suitable here if the rods are a tight fit in the holes.
If you wanted the handles to be removable for some reason (I can't think why) you can fix something like this in place with what are called grub screws, but they would be plainly visible on the underside if they were used and if made from steel there would be the potential for them to rust, even if zinc plated.