If you make a patch of the same species and close to the same grain orientation, it will shrink and expand exactly with the rest of the door. You would need a router and some chisels, a saw, a plane and some sandpaper.
Rout out a grave for a 1/4" or 3/8" thick patch so there is a larger gluing area and the edge of the patch makes contact on it's back corners all around. The grave can be about 3/8" bigger on all sides than the hole. A router with a mortise bit will give you a grave of reliably even depth all around, but you may want to use a guide to get the edges straight and parallel. You can square off the corners with a chisel, or round over the corners of the patch.
Cut the patch for a friction fit, apply the glue, and pound it in with a mallet and a block of wood, not too hard to split the patch. Repeat on the other side. If you want, you can cut a coarser plug of wood to fill in the volume inside and attach that before or after cutting the graves.
Plane and sand the patch perfectly flush, without going into the original material because a dish will be obvious when painted with shiny paint or varnish.
Casey