I vs'd a Swamp Thing with the item that heals him 25% hp every round. My shatter was at the back of my deck, so I had to turtle. Eventually he got to 80+ hp. How do you even deal with that?
There is always the option to ignore a champ and kill everything around it. I know it sounds silly at first, but it works and i've seen it happen a lot of times when watching others play. Like if its just a really tanky champion that deals moderate damage that you actually can ignore for 6 turns. Fade, ret are usually ur best bets otherwise.
Loss of Life *cough*, Essence Drain, champs with 7 speed that attack for 25% *cough* Damn im inflicted with disease and can't heal!
UD has many ways to kill high HP champs, usually 1 hit from another champ to soften, then Sheoul Demon + Mandate double hit does the trick, for example. The best part is you only need 4ap to double hit with him, which is pretty difficult to avoid. An aggressive UD BG can obtain board advantage relatively quick because of their damage output. It's my favorite way of playing UD personally
Debuffs. Hex, Stall, Sunder, etc... Make their tank a useless meat shield and then char broil that burger.