How to remove broken shutter fasteners from brick?
Shutter-lok fasteners are just made out of plastic (polypropylene), and typically anchor into a 1/4" predrilled hole.  They should not be hard at all to drill out with a sharp 1/4" drill bit.

An alternative is to:


Find a long metal screw, and pick a drill bit appropriate for pre-drilling that screw.
Drill into the remaining shank of the Shutter-lok, all the way to its end.
Screw the long metal screw into the Shutter-lok.
Grab the metal screw with locking pliers.
Pull the metal screw out.  (Ease it out, rocking and twisting if it resists.)


If you drilled the metal screw far enough into the Shutter-lok, they will come out together.  If any Shutter-lok plastic remains it will now be even easier to drill out with a 1/4" bit.