Over the years I've tried several approaches to reinstalling the shutters, many of which required getting the remnants of the previous plastic anchor out of the hole. It is tough to get leverage to drill into the wall while perched atop a 25 foot ladder and can be dangerous if you push too hard and tip the ladder over. The fastening methods I've tried previously didn't last, were too expensive or were too permanent. (If the shutter itself breaks, you need to be able to get the anchors out!) I've finally found a method that meets all my criteria and is easy to perform at high altitudes.
Step 1: Supplies
1) Pocket hole screws
2) Fender washers
3) Spray paint to match the shutters (not pictured)