Hi Nick - a few observations for you and three potential recommendations. TL;DR: go with an open-box or used pair of Focal Clear! I’ll explain below.
The observations first: typically, closed-back headphones offer inferior sound quality to their open-back counterparts because of things like internal cup reverberations and reflections as well as a sense of congestion, sometimes. The “inside the head” thing is worse with closed backs. Some folks, like me, also find them fatiguing because of the pressure that builds up inside the cup. Unless you really need the isolation or thicker bass response, I’d avoid closed backs.
I’d recommend some kind of open-back headphone instead, then, especially as you like classical music. Open-back headphones will give you the sense of air and space that you seem to prefer (at least as far as headphones can). There aren’t that many headphones in the $500-$1000 range. There are the Dan Clark Audio Aeon 2 headphones but I’m not sure how well they’d suit classical music. While I’ve not heard these latest ones, I wasn’t keen on the previous version as I found it to be too thick, congested, and muffled (which might be a function of being underpowered). See what others say. There are also HiFiMAN offerings but I’m unwilling to recommend them because of quality control issues. I have a pair of Audeze LCD2-C and am again reluctant to recommend them; I’m not sure I truly like them and, regardless, they’re not an all-rounder by any stretch of the imagination (they’re said to be better served as a rock headphone). The LCD-X might work, if it goes on sale or you could get an open box pair (see the reviews and impressions first, though!). The Drop Koss ESP/95X might also be a decent option, but its probably better for classical than rock (again, see what others say; I’ve not heard it).
Now for some recommendations. First, you might consider the Focal Elex. It is highly praised and might be just what you’re looking for - I’ve not heard it myself, though.
Second, you could go with the Drop x Sennheiser HD 6XX. Usually you’d want to spend the most on the transducer, then the amp, and then the DAC. But the HD 6XX is famous for scaling well with a better setup (so much so that some people drive this $220 headphone with $2,000+ amps). So, you could allocated the $220 for the headphones and with the rest of your grand get a nice tube amp - since this is something you’ve mentioned as a preference - and a complementary DAC.
One option would be the Schiit Audio Valhalla 2 ($350) with a rich-sounding DAC, like the Modi Multibit. That setup would set you back $820 before taxes and shipping. Another option from Schiit would be the slightly more expensive Lyr 3 ($500), which is a tube hybrid like your CTH (I assume you’re referring to the Massrop/Drop CTH here) with the same Modi Multibit DAC, either as a standalone unit or as an internal DAC card within the amp (the separate DAC is said to be better).
Alternatively, you could get a warmer amp, such as the ZDT Jr., and a more neutral DAC for the HD 6XX, like the Modius (I’ve not heard this DAC, it’s only just come out). The ZDT Jr used would be my recommendation for a warmer amp; if you go this route, make sure to get one with the hum issue fixed (it was an issue with the first batch but Drop resolved those that were sent back in, in which case the unit will have a white dot on the back, next to the serial number). I love mine. It’s great with the HD 6XX. But it needs a neutral or bright DAC (warm headphone + warm amp and warm or dark DAC would be too much!).
My strongest recommendation, though, is this: if you don’t mind buying open-box or second-hand you could get the Focal Clear. I got mine for well under a grand. (Make sure, if you go with used, that the pads and the underside of the headband aren’t discolored, which can happen all to easily; insist on seeing good pics first!). It’s an excellent headphone. I prefer it to my HD 6XX with my setup as it’s livelier, clearer, has better, tighter and deeper bass, and better treble extension. It’s more fatiguing than the HD 6XX, which isn’t saying much - it’s not a particularly fatiguing headphone, I can listen to mine for hours on end. It also has wonderful resolution and macrodynamics. In short, it’s got a lovely balanced, fairly neutral, well-extended, energetic and fast, clean sound. It’s ideal as an all-rounder, for me. It doesn’t have a particularly wide or spacious soundstage, but the imagining and layering are good enough not to make this as significant an issue for me (and I say this as someone who loves his HD 800 for its excellent soundstage).
Best of all for the Clear: it sounds wonderful out of your Drop CTH amp. That’s what I use. I’m toying with the idea of upgrading to the Lyr 3 with it. See what @lost33 has to say about the pairing, he’s likewise very fond of the Clear - as are a good many others on this forum. But the CTH and Clear pairing is widely praised as being synergistic (see here, for instance).
(Mods may wish to move this discussion to the general advice thread; these two threads here and here may also be of interest.)