REVEAL THE ANSWER
Start both the timers together and pop the eggs into the boiling water. After seven minutes, when the smaller timer runs out, restart it. After eleven minutes, when the larger timer runs out, and the smaller timer has been running for four minutes since it was restarted, invert the smaller timer. It will then run for four more minutes, running out after 15 minutes. All told, then, I only have to spend 15 minutes to get 15 minute eggs.