When Using Throne room on a Throne room do you need two more action cards already in hand?
In Dominion, the general rule is that you have to do everything on a card in order before you can play another card.

This affects Throne Room in an odd way, because it means you have to do everything printed on the card Throne Room tells you to play, then do everything on that card again.

Thus, when you Throne Room a Throne Room, you play one card and resolve it twice, then play a second card and resolve it twice.

TL;DR Yes, you can play the Laboratory you drew with the Moat as the second Throne Room's second play.