Lieutenant Tavson's pilot ability reads as

"After you suffer damage, you may spend 1 [charge] to perform an action. "

Having two charges, he should be able to recover one shield after suffering damage in two consecutive attacks during a round, right?

Yup.

Of note, this also means that if he drives over a rock, and takes damage from it, he can get an action, which is neat.

... or after taking damage from a bomb. Deciding where to reinforce after everybody has moved is also neat 😀