Since it was commented about and I think they're good ideas, I'll mention 2 things regarding specifics for my asking:
1. For price range, if it's worth it, I'm willing to spend. My hard limit is probably in the realm of 1.5-2k, but I also don't mind purchasing used items, so take that for a "real purchase price" rather than an MSRP.
2. I currently use a D50S/A50S stack which has served me well in the last 2 years I've used them, but there have been mentions of things that require some nicer amplifiers. Feel free to mention these! An example I saw was the HD800S that required nicer amplification. If your recommendation requires something like this, feel free to mention it. My 1.5-2k mention earlier only accounts for the headphones, so don't worry about that when it comes to amps/dacs, just be reasonable (i.e. recommending a 10k dac and a 15k amp seems a bit strange for a 1.5-2k piece of gear)
ok, you have a great budget,
There are a lot of options in this budget, I think hifiman arya v3 is an excellent option for classical music, the hd800 has a little more stage width but the arya has a wonderful and big sound, very detailed with excellent dynamics, the sound is alive and natural, it puts you in the atmosphere of the music,
If you want something cheaper but definitely very interesting, you can check out hifiman he6 se v2 (ADORAMA), these are headphones with supreme detail and excellent sound at a really cheap price, they have a different sound signature than arya (the arya with a bigger sound, the he6se with a better sound detailed) and they need a strong amplifier,
Meze 109 pro is also a very good option, although the sound is more fun and warm (but without sacrificing much technicality)
The question is also what are your listening preferences (are you looking more to analyze the sound and find micro details, etc. or are you looking for more warmth and fun, tonality, etc.)
Regarding an amplifier, ferrum oor with hypsos is a great option, it looks good, small, and it produces a great sound (it also has a lot of power)
Edit: Regarding iem, kse1500, I have never heard anything that sounds better with classical music (on iem)