I just posted in the PEE51 thread earlier, I picked one of them up today. Only had an afternoon of listening so far but it's quite impressive. I'm using it with my laptop, so can't comment on how much battery drain there would be on a phone (I have an iPhone and the PEE51 doesn't work with it). I haven't heard the Eagle though so can't offer any comparisons on that front.
The Chronos I've had for a few weeks has a slightly thinner sound than the PEE51, but having said that once I let my mind adjust to it, the Chronos is fine and doesn't seem thin over longer term listening. The Chronos has more mid-lower treble emphasis, and the impression of a bass hump - I say impression as it's not as if the frequency is playing louder, just that it's tuned to have a decent amount of punch and hit in a mid-upper bass region, which tends to make you feel or notice that part of the frequency spectrum more. It's also quite emotive and every now and then I'll find myself enjoying some particular nuance in the harmony or melody.
The PEE51 on the other hand has a more even approach across the bass spectrum, and feels like it has a fuller sound with more bass presence or reach down low. Still has plenty of punch too. Chronos itself has great dimensionality and sense of space, but I think the PEE51 beats it; really nice placement of instruments that helps keep everything distinct and separated. Overall the PEE51 sounds smooth but still detailed. Quite impressive what they've done with it, and it's cheaper than the Chronos too - but with less support for different devices. I think it will probably have more current draw and drain more battery than Chronos though, although don't have any way to test this directly (I'm guessing the PEE51 exceeds the iPhone threshold for device current draw, which is why it isn't marketed as supporting iPhone use). The PEE51 does get a little warm, but not bad at all even after a few hours on end; certainly less than Mojo if that's any use as a comparison.
Anyway, which one is necessarily best? I don't know, it seems after some time in this hobby all it really comes down to is things sound different, and which sound signature someone likes better is usually a matter preference.