I managed to beat (Advanced) Chairman @ Rook City on my first try.
Here's (roughly) how:
Team: Legacy/Argent Adept/Ra/Visionary
For about the first half of the game, legacy was using a galvanize/motivational charge combo, eventually boosted by Inspiring presence.
Later on, he got out a lead from the front, superhuman durability and next evolution.
Argent Adept did normal stuff for him, though what ended up helping the most was stack-playing the draw 5, hand/trash/etc. (Arcane Cadence I think). Letting Legacy use a 2nd next evolution every round was pretty darn useful too.
Ra... umm... did lots of damage.
Visionary ended up doing most of the work though, got a couple early brain burns out thanks to adept's vernal sonata (first one burned her, 2nd one was redirected to a decoy). I had her do a tiny bit of damage at least once too (psychic maelstrom), and lots of card draws to legacy. Oh, and having an ambitious racketeer shoot Operative a few times was nice too.
Oddly enough, rook city was what ended up winning it for me. About halfway through the game, I got out a Dr. Tremata, who got a Tony Taurus two turns later. Tony let me avoid a couple of Jailbreaks, which pretty much saved the game for me.
Final HP Totals:
Legacy: 4/AA: 2/Ra:4/Visionary: 3
First post Btw. Didn't do a story this time, but I think I will next time I post something like this. Gave me lotsa confidence about strategies. Not taking damaged thanks to legacy seems a bit OP imo though? Maybe next time I'll try without him.
(Confession: It wasn't the first time I fought chairman, I fought him earlier today with Fixer/Wraith/Expat/Nightmist, not on advanced... got wrecked)