How do I fix a door frame that my dog chewed?
You can simply replace the molding. This profile is an inexpensive ranch casing, available from your local home center. You should be able to match it virtually exactly. It will be nailed to the wall. Buy a new piece to fit. You will need to miter the top end to fit with a miter saw, but you can buy an inexpensive miter box to do that task accurately.

Once you have cut the new piece to fit, take the old one off and bring it into any good paint store. They can match the paint exactly. A quart of paint will be enough to repaint the molding with lots to spare for touch ups.

Install the molding using finish nails, set into place using a nail set. Then fill the heads with putty and paint.