This is an internet myth.
Dan never recommended any particular power requirement.
What occurred is that someone mentioned that the amps they observed in use by MrSpeakers were all at least one watt. In typical human fashion, someone processed this through their brain into "Dan recommends one watt".
Modern life is very complicated, and people are understandably desperate for any sort of simplification. These power ratings are just the sort of oversimplification that people love, but fail to adequately describe the situation.
For example, all of these vintage amplifiers output power like 20W, 65W, 220W to speakers. But to make those levels useful at the headphone jack output, there are resistors in series, so that it is possible to listen without having the volume control all the way down (!). The mathematical power at these jacks is usually about 250milliwatts, yet these amplifiers provide headphones with all the sonic benefits of a large amount of power, especially dynamics and bass quality.
By the way, the official word is:
You can drive the Mad Dog off your iPhone, iPad or almost any other quality portable media device (note that some Android phones won't cut it at all). That said, to get the most out of your headphones MrSpeakers recommends a good quality separate headphone amplifier and if possible a qualtiy source/DAC.