I was making about a dozen hard boiled eggs last night and a few of them cracked when boiling. The yolk partially came out of the shell but I continued to cook them as other ones in the pot were fine. Is it safe to consume the partially broken eggs after they cool?
Yes, it certainly is safe to eat them, it happens all the time. They won't last quite as long in the fridge as ones which don't crack, but as long as you eat them in a couple of days you are fine. They may look like nuclear mutants but they will taste the same.
To avoid cracked eggs heat the water more gently so they come to a boil slower, heating too quickly can cause the eggs to crack.