I can offer only a very limited perspective. I started with entry-level DACs, including one by Schiit, then upgraded to the Airist R-2R DAC, which is no longer in production. At the time of its release, it was an appealing DAC because it used an R2R implementation, which differed from the others out there (e.g. Delta Sigma DACs). It was also priced slight higher than the Schiit Modi Multibit and, according to Torq and some other reviewers, it was the best DAC you could get until you reached a much higher price range, around $1K. All this was before the Bifrost 2 was released.
The Bifrost 2 dramatically altered the cost-benefit proposition of DACs. It came very close to some of Schiit’s higher-tier DACs. Schiit, commendably, is not reluctant to release new and improved products even if they cannibalize the sales of their own higher-tier offerings. In this case, the Bifrost 2 rivaled the Gungnir Multibit DAC in some respects. This thread on another forum is enlightening about this, especially this post. I’d recommend checking out the various impressions there and this set of impressions, in particular, which for the most part align with my own (where they don’t it’s simply because I lack that reviewer’s experience and knowledge).
Schiit has a deserved, strong reputation for its DACs. The other ones mentioned in the posts @generic provided links to are also well worth considering, I should think, given the knowledge of those posting their impressions, and I’d also recommend looking into the Soekris DACs, which also receive widespread praise.
I’d recommend that you skip over the entry-level gear. Your budget allows you to do so, and you have a pair of top of the line headphones that warrant (and will do justice to) better-quality source gear. I’d say the Modi Multibit is about as low as you should consider going. It’s supposed to be an excellent DAC, slightly dark in its tonality, and still very good for the price. It’s been out for a while now, though, and is a bit long in the tooth, and I vaguely remember Schiit mentioning that they were planning to discontinue it before a fire broke out at the AKM chip factory, which brought a halt to the production of chips used in many other DACs. The Bifrost 2 is a meaningful step up from the Modi MB, and there aren’t many other DACs, from what I gather, that improve upon it significantly until you get to the $1K threshold.
Given your overall budget, I’d say allocating $700 on a DAC and the rest (some $1K right?), on an amp, makes good sense. The general rule of thumb is that headphones provide the greatest impact on overall sound quality of a system, then the amp, then the DAC, and finally the source. Only when people reach the top-tier of the hobby do sources and DACs become particularly important.
Finally, take a look at this excellent post from @ProfFalkin, one of our senior members and one of the most experienced folks in the hobby. He mentions the Schiit Gungnir A2 in place of the Bifrost 2 for the Vérité, which I suspect is because of the particular qualities of its pairing with the specific amp he recommends, too. I’m not sure if the Gungnir A2 is the current production one. I’ve lost track!