Most organic / real / analogue sounding DAC (ideally with Volume Control) for $1500??????

Chord TT2(with filter set to 4).  M-Scaler if you want even more real sound.
Digital cables do matter-I use Danacable and QED.
The romantic sounding tube amps are must for digital.
And speakers of course-I founded Proacs are the best,but Harbeth,Spendors,Stirling and
Sonus Fabers are options too.
And cables of course-copper Cardas,Purist,Kimber.
Only system approach works...

Sounds like you want to check out Lampizator. Tube DACs that were developed with exactly what you are describing in mind. For your budget you could pick up a lower end model, or something a little better second hand. 

I love my RME ADI-2 DAC. it is about $1,200.00. The best thing about it is it has a fully adjustable equalizer you can adjust to any room. Also has great headphone amp. I have a Schiit Magni headphone amp that I just quit using. The only bad thing is that it has so many variable settings. It took me months to get it dialed in the way I like it. Once you do, it's really good. A good friend of mine has a Chord Hugo 2 and is really thinking about making a change. He brought his Focal Stellia headphones over and was shocked how good they sounded with the ADI. I have Klipsch HP-3's that also sound great.

Aqua La Voce - a bit above your budget, but worth considering. Beautiful analogue sound made in Italy :)

I have been a digital refusenik for decades, fully believing that nothing could ever come close to the experience of listening to vinyl  on a decent system. For me, digital was just a matter of convenience — for background listening on a Sonos system. But when I wanted to really immerse myself in a recording, it had to be vinyl. A few months ago, my analogue world was rocked: I decided to improve my Sonos streaming experience by getting a Wyred4Sound modified Port and a W4S DAC 2v2SE. All I can say is: I was wrong about digital. Yes, playing original vinyl  from the pre-digital era still sounds better than the digital versions, but that has to do with the often-inferior digital masters that are used as source. Most of the time, doing A/B tests there is little or no  difference — as long as I’m streaming CD quality — eg the “new” Neil Young record: if anyone was going to be careful about how his music is distributed, it’s Neil. The album was recorded and mastered in the seventies - so pure analogue. But the digital and vinyl versions are completely indistinguishable on my system. So one possibility, while remaining in your price range would be the lowest priced W4S DAC: the company provides an amazing upgrade path should you wish to spend more at a future date, without having to start with an entirely new DAC.