Is there a special way of discarding leftover bread/challah?
Feed it to the birds, or wrap it in a bag prior to throwing into the garbage.
1) Eat it as is.
2) Eat it in some concoction. French toast is one possibility.
3) Why do you have leftovers in the first place? I think people should not buy excessive quantities that tend to can lead to such a situation ab initio (lechatechila). Doing so could possibly be a gram of bal tashchis (problem of being cause of wasting food indirectly).
4) Do not dump them on someone else's property or on public property to allegedly feed birds. Other people do not necessarily want pigeon encampments and what they leave behind. If you choose to do that, do it only on your own property.