I would have to completely agree with Project86 (Project86, get a set of balanced T1s NOW). I have a few friends, including myself, that own/owned pretty much ALL the top of the line headphones and we are ALWAYS reaching for our T1's lol (this also includes some people from highly respected aftermarket HeadFi shops, yes plural). We might have several DACs, but we only have one set of ears hehe. Anyways, just about anyone on this level of equipment will first off tell you to go upgrade your cable to balanced. Moon Audio (no affiliation) can do this for you with their Black Dragon cable with 6-10ft in the $250-300 range with round trip shipping.
Ok, now for the DACs. Again, friends and myself with this level of gear are currently highly recommending the Violectric DAC/amp combo from AphroditeCu29 (no affiliation) as a "goto" DAC for its all around accuracy and transparency. With that said, there's a new guy in town, Yulong's flagship D18/A18 combo. I have heard the Violectric myself, but NOT Yulong. Given the experience of ZombieX and Project86 here, I do trust what they have to say and will I be buying the Yulong combo since it pretty much makes ALL music sound good, judging by their descriptions. The Violectric is so transparent that some bad recordings will be unbearable to you. Another thing that tips the scales in Yulong's favor is if you're looking to buy right now (1-6 months), because Violectric is currently working on a prioritized XLR (true balance, like Yulong) DAC and Amp, but release date is uncertain.
Either way, the Violectric for transparency, or the budget conscious Yulong with a vinyl touch will be excellent choices. Just be sure to recable those T1s over anything else, and they will be your go-to headphones over HiFiMan, Sennheiser, and Audeze. Just like my friends and I, along with some HeadFi'ers are doing once you're this far up the ladder.
P.S. I personally have a problem spending over $1k on DACs as they are only in their infancy right now, and are changing quicker than ever before. A "better" DAC is always on the horizon, and I'm sure some people that spent well over $1k a couple years ago are looking to upgrade (secretly of course lol), or have already done so.
Hope that helps and enjoy!