“Dès que” means “as soon as” and is used to express that one future action will happen after another has been completed.
The main verb “corrigera” is in the futur simple, which tells us the letter will be corrected in the future.
But before she can correct it, the letter must be written. So the writing happens first.
To express an action that will be completed before another future action, French uses the futur antérieur (future perfect).
“Tu” + futur antérieur of “écrire” = auras écrit.
So the complete sentence is: Dès que tu auras écrit la lettre, Mme Dupont la corrigera.