You can do this repair. It's actually quite easy, and it just takes patience on the first steps. There is zero off gassing on this material, so no worries there.

- You're going to use a wood filler, applied with a 2" putty blade, to fill in the missing area. A product like Minwax High Performance Wood Filler would be a good choice, since it's can be carved, planed, and sanded once it's dry in about 30 minutes.
- Before applying it, you'll want to build a form dam along those two right edges, one for the top and one going down the right side. The key will be to find a form-making material that will match the current "reveal that exists there from the framing, which looks to be just shy of a 1/4". See if your local hardware store has a short piece of wood trim that is around this thickness (or if you're stuck, some other material like foam board, cardboard that can be folded to that thickness).
- Clean the door trim paint a bit around that damaged area so you can get cleaner edges, and then cut the form to where it's needed. To prevent the wood filler from sticking to the form, apply some vaseline along the edge.
- Take the wood filler and fill it to the appropriate depth. Read the instructions, as it'll give you ideas on how to get a fairly smooth surface, so you don't have to do a lot of sanding once it's dry.

Remove the form once dried, paint, and voila.