You can’t wake up a deep sleep device with mqtt. It would need to be awake to receive the message. You can do it with a wire… But doesn’t really help you(well perhaps the boiler could control that actually)
You can enable/disable DeepSleep and adjust the DeepSleepTime intervals, while the device is awake.
When the boiler is going you could send a message to disable DeepSleep, the esp8266/esp32 would need to be waking up to act on that message though.
Big enough battery/solar panel setup could get you a decent runtime…
I have 4 18650s and a small solar panel on my esp8266 roof temp sensor. It only wakes up every 10mins and sleeps at night though.
Esp32 have better low power management I believe.