Pardon the mess. I wrote this with a different markup tool and it looks like it doesn't want to play nice with Ghost.
I started writing this a few days ago, and have since ironed out some things which I hope to come back and flesh out. I leave the rest of the post as-is for now, half for my Dear Reader, and half as a note to self.
TL;DR: I'm virtually positive I'm right about most of this, and have come up with several different ways to theoretically exploit continued fractions to generate primes of arbitrary size. Unfortunately, and naturally, PCFs are just as deeply nuanced as most prime-connected stuff, meaning same ol' story, different tune.
If it were possible to cheaply generate large PCFs, it would be a very different story, and I haven't completely ruled out the possibility yet. In fact, it would be sufficient merely to develop an algorithm which could spit out the length of the period of the PCF for a given radical, assuming it ran in reasonable time. What I did find in the literature about it suggests that it's an open problem, but there also wasn't very much literature compared to some of your more mainstream areas of number theory.