Pulled broken glass out of my tire . . .

[NbyNW]

Thanks Eden! Glad to know it's nothing to worry about . . .

. . . and kind of surprised I've managed to avoid this happening until now.

[RoadRaven]

If you have a cut and the cut goes right through the tyre, I assume you had to change the tube?
When you change the tube put a piece of old tyre on the inside of your tyre against the cut before you replace the tube.
This will preclude any bits of stick, glass or other irritants working their way through and giving you another flat.

If the cut doesn't go all the way through, just check each time before you ride, and if these were race tyres, replace the tyre and use it on a training bike.