I am doing similar right now. Using Formica 459-90 Brite white laminate. As already mentioned, the way you handle it is by applying edge-banding to the sides of the panel. I wanted the exact same shade of gloss white on the edge-banding and had a very hard time finding it. Only thing I could find to exactly match my Formica Brite White was at Cabinetmaker Warehouse they have a (pretty expensive) 600 foot roll of unglued edge-banding. When I got it, I looked at the packing slip and realized it was just generic gloss white. (although it does match pretty well)
An edge-banding machine would be nice to have, but you or I don't have one of those. If you find pre-glued, that is probably best. I have unglued. I think most would use hot-melt to apply it, but I am using contact cement. They make trimming tools for edge-banding, but I have some utility knife blades that are beveled on one side only, and just use one of those in my hand.
Good luck with your project.
Actually, I am going downstairs to put banding around a couple of doors right now.