Monastery is a far more powerful trasher than it may appear to be at first glance. An obvious point of comparison is to how Forager plays in the early game. Like Forager, Monastery does not take an Action, trashes Estates for no (other) benefit, and trashes Coppers without hurting your economy for that turn. Where Monastery shines is that it allows you to trash without sacrificing any momentum (beyond drawing the Monastery) — you don’t give up economy, you don’t have to worry about drawing it dead, and it even trashes faster when used in tandem with gainers or +Buy.
Monastery’s properties allow for more aggressive openings than one would typically do with a trasher. Consider an opening of Smithy / Forager. Normally this is a bit risky – the Smithy could dead draw the Forager, wasting a shuffle’s worth of trashing. Since Monastery is played after the Buy phase, it can’t be drawn dead like Forager in this situation. In addition, you get all of the benefits of Smithy’s extra cycling and economy, leading to more frequent trashing and faster gains. This can quickly snowball into a very potent deck that trashes seemingly without effort.
The synergy with gainers such as Ironworks or Engineer are obvious – gain a card, ideally buy another card, use Monastery to trash twice as much as you normally could. But this synergy often enables even weaker cards such as Silver gainers which you may not otherwise consider. Squire is an interesting example, where it can gain Silvers early to give Monastery an extra trash, provide multiple Buys later for a similar purpose, and can even be turned into an Attack card if need be.
In games where you need to get very thin very quickly, opening two copies of a trasher is often wise, but that is usually incorrect with Monastery because if two copies of Monastery collide, you may not be able to gain a desirable card which sacrifices momentum. You do often want two copies of Monastery, but try to stagger buying each Monastery with buying more aggressive cards that draw or gain to keep things moving. The main takeaway from this is that you should adjust how you pace your deck when using Monastery instead of other trashers. Play to the strengths of Monastery, and you will be very glad you picked up this unique trashing card.
For more detailed discussion, you are welcome to visit the Dominion Strategy Forum thread about this article.