Probably more than you want to pay. There's no way to patch the area immediately around the hinge; a repair is going to involve cutting off perhaps 2" of that last board, along the entire length of the door. Then you've got to obtain a similar piece of wood (not sure if it's pine or cedar), glue it onto the cut edge, sand the joint smooth and flush, machine a rabbet on the back, round the front edge, and completely strip and refinish the entire door before reinstalling it. Besides time, there are the costs of lumber, finish and possibly a specialty router bit.

If you're not concerned with how it looks and only want the door to open and close then perhaps you could stop short of the refinishing. It would look like hell, but it would function. But to do a clean, professional job of it all the way through is not a quick and easy sort of thing. If you're imagining the cost should be in two digits, you're going to be disappointed.

If the broken piece is still in one piece and is available, repair should be simpler. If the piece is lost or really splintered, see answer above.

Another option is to have two functional doors made. Doing two of the same thing is seldom twice as much effort as doing one. And sometimes making a replacement is quicker than making a repair and trying to match everything all up. This would eliminate the profile matching, thickness/joint matching, stripping, color matching, etc.

The wood is Aromatic Red Cedar. In fresh form, it is very pink/salmon color with white sapwood. Your door has gained a lot of patina from age and I have found this difficult to seamlessly color match because just as soon as you are done, the wood is starting to darken and you need to undershoot the color and hope that it matches in a few years. As is also typical with red cedar, the finish has a bit of cracquelure that is difficult to replicate. So the new section will have a smooth finish.

You could also consider making it from a paint-grade wood such as poplar and painting it a complementary color (black?)