The primary key to beating Tauntauns is cutting off their flow. This means that you will make sure they cannot use their abilities. If they cannot use their abilities, you can beat them with less difficulty simply because it does not matter how good the cards are when they are stuck in hand or when they cost 5 resources per card! You should be thinking about making them discard powerful stuff and rendering their high resource costs ineffective. Let's talk about some common losing scenarios and solutions.
1. No Answers
When you see your opponent committing too many resources into a single target, you know there is no option to stop you from breaking through. Your opponent might be likely trying to draw answers, but what happens when they won't find any? You can break the target and win the game! With that being said, a simple answer might just be a couple of immediate damage effects or threat reduction that can wreck Tauntauns before they have a chance to use their shenanigans.
2. Passive Play Yields The Field
This situation usually occurs because your opponent has committed too many resources into a single character, as well as the fact that your board state has been suppressed so much that it would cost many cards to take out all of those characters without suffering too much harm themselves. Now, since our characters are huge and we don't have a fast win condition, we have to take the game long and try to grind out 3 or more damage effects that can end the game. If your opponent is playing against you with Tauntauns, then chances are they will also be working towards this objective as well!
3. Threat Saturated
If you can get to a point where the only way your opponent can win is through Big Game Hunter, then you have effectively won. This means that you should be actively trying to avoid having too many characters on the table at once simply because your resources will go to waste if they are all killed off!
4. They Draw A Defensive Plot Twist
This situation occurs when your opponent is going for a slower strategy and decides to play defensive plots while not committing too many characters or upgrades in an attempt to beat you with incremental damage via Best Defense. This essentially means that you need to finish them off before The Best Defense comes into play (which usually involves getting rid of Tauntauns' partners in crime).