The first time a toilet is flushed in the morning, we get a loud 'bang' sound (very rude awakening!). We've isolated it (I think) to an outgoing problem by turning off the water supply to the toilet and then flushing it. What could be causing this?

It sounds like you might have a faulty ballcock. Do you get a "bang" when any other faucet is shut off? Regards, Tom

No, there's no feedback from any of the faucets (which rules out water hammer). And I don't think it's the ballcock, cause it happens on both the toilets, depending on which one gets flushed first in the morning, but I'm willing to test that idea. How would I go about it?

But there's a "bang". We install air chambers,( shock absorbers) to absorb "bangs". I know of nothing that would cause a bang in the water lines except water coming to a sudden stop. By the way are you on a pump or city water? If on a pump please describe the set up. Why not recharge your air chambers just to check it out. Back in the 60's we were required to to put shock absorbers,(air chambers) on all bath lavatorys and kitchen sinks. Over time water and condensate built up in them and they must be recharged. First turn off the house at the main house shut off valve. Then open hot and cold faucets at the farthest bathroom to prevent air lock.(This is important.) You will find under each fixture little brass or chrome valves. These are called angle stops. With a small pail held under the angle stop to catch the run off look under the handle of the stop and you will see a nut with the stem running through it. This is called the bonnet packing nut. Put your wrench on it and back it off counterclockwise. Now open the stop and remove the stem and washer assembly, The stop will begain to drain. When it has quit draining reassemble the stop and move on to the next one. Do this on each one of your angle stops. There will be six of them. When you are done, close off the faucets you have opened, turn on the water to the house, and give yourself a pat on the back.
You have just recharged your air chambers. Good luck, Tom

It would be nice to know when this started and how long it has been doing it. Is the first flush in the morning the only time it does the "bang" or does it do it sometime later after the commodes have not been flushed for several hours?
It also might help if you would try to describe the "Bang" does it sound like it is coming directly from the commode or does it sound all over the house.