I can't comment on either the Burson or the Shiit stack, but I've had the LCD-X for a few months now, and have enjoyed getting to know them on a couple solid state DAC/Amp combos.
The LCD-X are quite an interesting beast: one one hand it's insanely easy to drive (it has stupidly low power requirements -- you can run it off your iPhone and it sounds passable) -- but on the other hand it's also quite revealing. The source/amp combo on these cans is less about how they drive, and more about what character of the sound you want to experience.
One note of caution: the LCD-X are so sensitive, you can easily find yourself stuck in the first 1-2% zone of the volume knob on some headphone amps (like the Bryston) -- particularly when running them balanced. Make sure you can audition them with the amps you're looking for before you buy-in.
Compared to the Sennheiser HD600 and HD800s, the LCD-X are very bassy and slightly more "polite-sounding" (think less etched-sounding treble). The soundstage is smaller than the HD800s, but the powerful low-end tends to create a big envelope of sound (read: expansive sound, but it's harder to resolve specific instruments/locations)
I've been going back and forth between two setups (both in 4-pin XLR balanced mode):
- Benchmark DAC2 + Bryston BHA-1
- Oppo HA-1 (both DAC and amp)
The Benchmark/Bryston combination sounds fantastic. The only catch is that the LCD-X are just too sensitive for the Bryston amp. I have to pot the Benchmark DAC2 down from unity gain in order to listen to a reasonable sound level on the Bryston. The BHA-1 sound is fantastic on the LCD-X, but it doesn't seem well-calibrated to run extremely sensitive headphones.
The Oppo, on the other hand seems like it's in the goldilocks zone for the LCD-X. I'm not sure whether it's the DAC or the amp portion of the Oppo that has this character, but it's just slightly more detailed in the treble than the benchmark/bryston combo. This pairs really, really well with the LCD-X, making them feel slightly less polite and rolled-off in the top-end. The low-end is powerful, but what sounds like just a little more detail in the treble really tends to round out these orthos.
Verdict: the Oppo HA-1 is an amazing match with the LCD-X (it's also a fantastic, very high-quality all-in-one DAC/headphone amp). The Benchmark/Bryston combo seems to fit much better with the Sennheiser HD600, though -- they and the HD800 benefit massively from balanced operation. I'm still undecided which setup I prefer for the HD800, though.