Pick loose splintered wood off the door by hand. Sand the edges of the hole with 80-grit sandpaper until no longer jagged. Sand a 1-inch to 2-inch section around the hole to remove the existing finish.
Spray a paper towel or rag with an all-purpose cleaner and wipe the area around the hole to remove oils, dust and dirt.
Direct the spray from a can of spray-foam insulation into the hole and fill it until the spray foam expands to slightly higher than the exterior surface of the lauan door. Let the foam insulation dry until it is hard, usually 30 to 60 minutes.
Insert a straightedge razor blade into a scraper and shave the excess spray foam off the surface, making the insulation flush with the door.
Scoop up premixed drywall joint compound on a thin putty knife. Spread a 1/8-inch layer of joint compound over the insulation and around the edge, feathering the edges 1 inch to 2 inches onto the door as you move away from the repair onto the lauan. Let the joint compound dry for three to four hours.
Wrap 180-grit sandpaper around a sanding block and sand the surface of the joint compound until it is smooth and flush with the surface of the door.
Wipe the face of the door with a tack rag to remove the sanding dust.
Apply a coat of interior-grade latex primer over the repair with a paintbrush or roller. Allow the primer to dry for three to four hours. Apply two coats of interior-grade latex paint over the patch, allowing each coat to dry for two to four hours between applications.