Vandene had a long conversation with Moiraine, and half-puzzled out many things that were happening (Forsaken loose, the Dragon being reborn etc.).

We were not privy to the conversation that they had after the Shadow nearly killed both of them. I imagine Vandene suddenly had a few pointy questions, and Moiraine didn't really have a choice to trust her and open up, at least a bit. Moiraine had put Vandene's life at risk by coming to her and not telling her about the potential danger she was in, after all. Later on, we saw Vandene finally able to convince Adeleas to come out of retirement (she spoke to Moiraine of the LB being imminent and wanting to drag her sister back to the Tower).

I'm pretty sure it's then that Vandene decided to teach Moiraine balefire, as it's one of two possible Moiraine episodes that fits, and Jordan liked to hide answers like that in plain sight and leave us to puzzle it out. (The other possibility is that Verin knew it and taught it to Moiraine in Falme... but it borders a betrayal of the DO a little too much). In any case, we have a timeline, based on Moiraine's answer to Perrin, confirmed by her failure to deal with Aginor and Belthamel. She didn't have balefire in EOTW, and in TDR she said she had become "a lot more dangerous".

I don't think that weave was ever "lost". I think above a certain rank in the Green Ajah hierarchy, the sisters are taught balefire to preserve knowledge of the weave. The Green Ajah might need all the destructive weaves still known one day, for the last battle - even that one. It's forbidden by Tower Law, so the Greens keep this secret, and it's very unlikely all but a handful of old Greens are passed this knowledge, and whenever one of them dies or retire, a new "guardian" is added, so knowledge of these weaves don't vanish. It sounds like the sort of weave the Captain-General would trust only the most level-headed sisters with - sisters like Vandene, and Cadsuane (who recognized balefire on sight when she saw it...) etc. Greens that will never use it - and that they won't reveal to others they have before the LB, and then only if it's necessary to defeat the Shadow. Primarly, they're just repositories of dangerous and nasty knowledge the Green Ajah can neither afford to use nor to lose, whatever the other Ajahs think. My guess is that most of the Green Ajah's "secret weaves" is nasty stuff like that, balefire most likely the worst and most restricted one of the lot, while other deadly weaves are probably taught upon gaining the shawl. Cadsuane might even be an exception in both knowing the weave and having the strength to make it work without an angreal. Vandene might have taught it to Moiraine the way Sorilea gave Travelling to Cadsuane.

As we saw in the Two Rivers and at the Cleansing, the Green Ajah is arrogant, remote and a bit aloof, but when faced with the Shadow, even a fairly young sister like Alanna (and her warder) has tricks up her sleeve: knowledge of strategies and tactics, weaves two sisters can reasonnably use without exhausting themselves completely, knowledge of weaponry etc. One even wonders how much theorical knowledge of things like artillery the Green Ajah/WT still possess from books and doesn't share to non-AS as it would make warfare more dangerous as well as reduce the superiority of the WT over the nations (and even threaten their policy of non-involvement.)