The BH just gets out of the way and lets the phones do their thing. Not only from a fully DC-coupled neutrality standpoint but by supplying the phones with all the current they need, no matter the load. At first glance this appears to be a boost to the bass region and some extra treble but change the fit to make the bass a bit leaner and it all snaps into place. The BH will never be an amp to impress and it gives ruthlessly revealing a whole new meaning.
The ES-1 is a very limited from a design standpoint and no amount of bling caps and other crap like that will fix it. It is after all just a 1968 Stax design, a good one though but it lacks all the refinements that the last 40 years have brought us. Then you add the crap construction and the possible filament fiasco and amp will chew through those expensive tubes in no time. Before you commit to anything but new production tubes in the ES-1 I would have it checked out to see if the same filament wiring is feeding the entire amp. I've heard enough horror stories...
That said you might prefer the ES-1, hell you liked the Jade after all and the ability tube roll may be appealing to some. With the old BH I tried every EL34 I could get my hands on but after that the Mullard XF2's were the only ones I used and a single quad at that.