Tort beak is kinda like fingernails, it grows all the time but depending on living conditions and food it maybe more robust or brittle. It should grow back but it may need some extra attention now
If you're in the States and bought the tortoise from a pet store, chances are pretty good that your tortoise is older than three years - more likely about 10 years. And a tortoise at that age who hasn't been allowed to bite off his food from the stem it grows on, probably has an overgrown beak. In this case, it was the "overgrown" part that broke off. When a beak is allowed to get overgrown, and it breaks off on its own, it sometimes breaks off much shorter than it should, and this is very painful.
You can see I'm doing a lot of guessing and surmising. We'll need to see pictures to give you better info.