Maybe it breaks because it is not properly cured or because we forgot to use Oven bake clay adhesive to adhere raw clay on to baked.
I broke some pieces when I drilled them, they were layered so nowadays I use Bake and Bond more often.
Anyway! If you happen to break something chances are that if you press the 2 pieces together by hand the break is not visible.
But when you let go the break shows up, what to do? You can´t hold your hands in a hot oven for 30 min.
How to Save Your Broken Baked Clay
I say put the pieces in the oven for about 10 minutes just to get them hot, working fast you now apply Bake and Bond to the break and push the 2 pieces together and hold until that heat cures the Bake and Bond just enough to handle your now fixed piece. But BE CAREFUL it is hot and you can burn yourself.
Then bake as usual for full time and voila your piece is fixed!
Another way if you can´t handle the hot clay is to put a little Bake and Bond on both surfaces and a small piece of raw clay on one side and press it together, the raw clay will hold everything in place. With this method it is more difficult to get an invisible mend so you choose which method to use.
In the pictures you see a dragonfly that my daughter made from Pluffy and one wing broke off during play but we fixed it.