I don't know if head-audio works quite the same...personally, I've developed a bigger interest in many different types of headphones, as each pair has its own strengths, and the same applies to amplifiers.
But as for amps, Asgard is the solid state amp of Schiit's middle tier. Good on current capacity, less good on voltage (hence the lower power output at very high impedances). Valhalla is the all tube amp, with excellent voltage potential, but with less current output. Lyr is the best of both worlds. Using a hybrid design, it can source much more current than Asgard while still managing good voltage for high impedance cans. (Also note: you might find lower prices on used versiona of the original "1" models. The "2's" available straight from Schiit are a little better, but also include gain switches and preamp outs, offering more function over the older models.)
Anyway...I think Asgard will probably do well. Lyr will have some more versatility. My recommendation, however, is to try everything. You clearly indicate that you aren't on the verge of purchasing any of this tomorrow, so try to find some meetups or a real store to test these for yourself. If you can, get the headphones first and plug them into someone's amp (with premission!) and see how you like it.
Oh, and keep in mind that Schiit do offer 15-day satisfaction guarantees. So if you have a flexible bank account, you could try a couple amps and keep the one you prefer. You'll only lose the 5% restocking fee (plus shipping, maybe) on Lyr and Asgard.
