How do I fix a broken tile?
The tile broke because it was not supported, that is the hard part for you to fix. Remove the broken piece, and see if there is simply a gap underneath, or if there is no pad/foundation at all to build on. If there is nothing but dirt or air underneath, then you need mortar and and enough of a base that it ties into the existing pad and is also below the frost line. This will be hard to do properly without removing additional tiles. If there is just a gap and there is something directly below to support it, and if the gap is small enough, I would use a two part epoxy to repair. I know some brands in the dual tube packages say they work for masonry on the label, and I have used them in the past for similar repairs. Epoxy is stable, resilient, and thin enough that you wont hardly notice the crack. If you get freezing winters in your area make sure there is no place for water to get in and freeze or it will only break again.