Best Headphone system under $1000???

I really like my old Stax Electrostatic headphones and their amp to drive them.    I'll admit I am biased toward them, but it wouldn't hurt you to start your listening / shopping experience with them and move on from there.  

Do check out headphone systems at twice your budget so you know what you are missing out, if anything.   At least, if you do discover the difference, you can make an honest judgement. 

Be careful though, it is very common  to do permanent hearing damage with headphones as pumping too much SPL in the tiny ear canal is easy to do.
For open back headphones, in general if you can't carry on a conversation with someone in a normal voice at 3 to 4 feet, the phone volume is too loud.  Turn it down!

I'm not sure if it's reasonable to expect to "replicate" what you have in a headphone system, the experience of listening to music through speakers and headphones is very different.  Your budget allows for some decent components. 

If you're willing to buy used, consider the Audeze LCD-2.  They are detailed, but on the warm side and have good bass and will sound great with the types of music you enjoy.  You can get some new earpads for a set of used of headphones if you're squeamish about buying used.  The Audeze ones are about $100, there are some nice aftermarket ones out there for less.

If you prefer new, check out the Hifiman HE-560.  They are similar to the LCD-2, but are harder to drive and will require a more powerful headphone amp.  Another great set of headphones is the Focal Elear (or Elex from Massdrop).  These are more "neutral" than the other two I've mentioned, but are very detailed and easy to listen to.

An outstanding headphone value that I recommend looking at is the Massdrop Sennheiser HD-6XX headphones.  These are super comfortable and sound great and are only $200.

Like speakers, the headphone is going to impart more to the sound signature than the amp, but proper amplification is important.  I don't have any suggestions in your budget range, most of my amps cost more than that.  If you can find a used  Musical Fidelity M1-HPA or M1-HPAP, that's an inexpensive amp that I have owned that has a lot of power and sounds good.

The system I use and can therefore recommend is Focal Elears and Schiit Jotunheim.

The system I use and can therefore recommend is Focal Elears and Schiit Jotunheim.

I was going to mention Schiit, but I'm not familiar with the models that would fit in the OP's budget.  I have a Mjolnir 1 and it's a great amp.  If the Jotunheim is close in terms of sound quality and performance, that's definitely a good option to consider.