There are a few I'd absolutely recommend without any hesitation.
One of the best,and most versatile would be the Sprout 100 (designed by an audio legends son, Scott McGowan, his father Paul is a family friend, so maybe a slight bias) but truly a extremely well rounded dac/amp. Many use it just for headphones but can easily be used as an all-in-one for a 2 channel setup as well. I've found that while it does easily power most PM headphones, the midrange can become a bit lost; but with dynamic headphones it's amazing. Obviously at the top end of your budget, but due to its versatility, it can really be a great product. Also handmade in CO.
Next I'd say the Audeze Deckard. Obviously an amp biased for its own headphones, but plenty of low end grunt and enough power to power the most difficult to drive cans. Extremely subtle but a very balanced and warm amp as well.
Arcams rHead is also one great amp and about half your budget, but there isn't a built in DAC with this one, but probably one of the best sounding amps for under 1k. It's a very clear and focused amp which I've found to be one of the more natural sounding ones around. In essence you hear what the artist intended you to hear. But some people don't like the lack of over accentuated lows to highs. More of an old school amp imho.
Burson Audio Conductor V2 is one helluva amp/dac. It has a Sabre32 DAC, which is pretty amazing. You should be able to find this in the States for under a grand.
Simaudio Moon Neo 230HAD another great versatile amp/DAC et cetera. Very linear and warm sounding, and an amazing low end. Not quite the clarity you'd expect at its price, but what it lacks in that aspect, dynamically is amazing. Now the last two I've mentioned would be pushing your budget, but easy to find on sale.
Schiit Audio Lyr 3 Hybrid with your choice of optional DAC (should come in around 1k) I've only listened to it a few times but was blown away. For the price probably once of the best hybrids out there.
Schiit Jotunheim Balanced amp with your choice of a DAC, Phono or Multibit DAC. Another amazing product from Schiit. Mike Moffat is what Rhamnetin thinks he is lol. He's a brilliant guy who can make these amps compete with ones costing 10 times what they cost. One great thing about Schiit is 98 percent of every component they use is made in the US. While many audio firms make their products in the US,not every single bit of say an amp is made in the US; a DAC may be made elsewhere, but Schiit is committed to everything locally sourced. And it shows. This is one of my favourite amps for under 5k. It's truly amazing. Punchy, transparent, dynamically sound, amazing build quality and an amazing warranty if that matters to you.
A Woo Audio WA7 is also a great tube amp,but I'd only recommend it if you can swing to also get the WA7tp PSU, as the standard PSU isn't the best and it does lack the deep lows you'd expect from a tube amp. The sound is "cooler" than most tube amps, which is a personal preference, but it's a great combo.
The Woo WA3 and WA6 are also excellent as well,minus the DAC, which may blow your budget. The oversound are very similar amongst the Woos, so it's all dependant on what you like. One issue is tube rolling is pretty much impossible with the Woo Audio tube amps, so if that was something you'd be into, scratch that off your list.
Little Dot, although Chinese do indeed make some decent gear for the price. I'd say if you wanted to start with a very inexpensive tube amp, and roll, and play around,I'd say to try one out. If you buy direct, Yang usually offers a lifetime warranty, only caveat is you must pay shipping back to Shenzen I believe after the first 2 years. But it's a definite showing of confidence in his products. And they are a tube rollers dream. Granted I wouldn't get anything below a MKIV, although Chinese made, their components are very high quality, and they do give mainstream manufacturers a run for their money. I have a Little Dot MK VIII SE which is a great little amp for under 1k, and I use it mainly to compare with other amps I've been buying, and it can definitely hold its own. Don't confuse it with a McIntosh MC275, but for its price, cool little thing.
Hope some of this helps, of course everyone has their own preferences and I'd suggest to find a good hifi dealer than has a generous return program, or even has loaner products you can try out and if they don't work for you, you can return them.