While the E12 will power your HD700's just fine and is a nice portable amp, if you want to get the full potential out of the 700's, a desktop amp is called for. I am running the Lyr2+HD700 combination for the last month and I had been running the Lyr for a year prior. I also have a very fine portable DAC/Amp in the Practical Devices XM6 that I have used with the 700's as well as time with the Asgard 2. All of these amps make the 700's sound very good with full bass thru treble extension. I haven't found any major frequency response variations among them when the Lyr or Lyr2 is running with its stock tubes. In fact if I had only ever hear the Lyr in this way I would have kept the Asgard 2 as it is quieter and $200 cheaper.
However, one hasn't really heard the Lyr or its new version unless you have tried it with premium NOS tubes! Where all the amps above present detailed, extended ear-to-ear reproduction, it is largely dimensionally flat in the plane of your head unless one is listening to binaural or sounded recordings. With the premium tubes that I have (Lorenz Stuttgart 6922, and Voshod Rockets 6N23P) the presentation changes dramatically. Quality stereo mic'd recordings that have an acoustic cause the presentation to expand forward of your head. In addition the instruments start appearing. What I mean by that is in "flat mode" you hear the notes, strings, skins, etc but, they are disembodied. With these tubes they link up so that you actually hear the instruments along with their music as a whole making the performance deeply satisfying in the way live music is. Now there are differences in the two tube sets and I found the Lorenz has a bit less extended treble compared to the VRs. But this is subtle compared to the change in dimensionality and coherence. I simply find the stock tubes too "in your head" and prefer a more conductor/audience presentation for dedicated listening sessions.
I know others here enjoy the new Valhalla 2 which can now accommodate the efficiency of the 700's. I have not heard that combination. I preferred the Lyr over the original Valhalla at the frequency extremes and do not feel the Class A MOSFET output stage imparts any grain or harshness. I come from using Stax for years, so am particularly sensitive to clean treble presentation. Finally, I do not have extended experience with other tube or hybrid amps, but I don't believe you would second guess yourself if you went with the Lyr 2.