I perfected my egg-cracking technique!
It just struck me yesterday morning as I was starting to crack six eggs.
I tapped the side against the bowl like normal, but then I took that broken side and turned it upward, pointing the egg fragments away from the bowl.
Then I dug my fingers into that break and cracked the shell downward and then horizontally, perpendicular to how it would have broken if I'd just pulled the egg open using that first break (I only opened it enough for the contents to come out).
Doing this snapped open the new "bottom" of the shell cleanly, every time.
I used to be paranoid about shell fragments and would search for them, but doing this seemed to prevent it from even being possible.