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You can save more energy probably if you only put a little bit of water, and then steam the eggs in the pot (no need for a steamer basket, just put the eggs in the shallow water). Also, apparently, peeling is easier when the eggs are placed in hot water vs cold. Look up Kenji's egg boiling experiment stuff.

I think you can make it even faster by electric kettle to give a little assist. Maybe replace 1/2 or 1/3 with boiled water. That starts you off with half boiled water so it comes up to boil faster.

I go back and forth between boiling the eggs the whole time, and bringing them to a boil and letting them sit. I can't determine which works better.