There is a Thunderbird setting for this Rated 5 out of 5 stars
A per-account setting enables sent email to be immediately enclosed in whatever folder instead of default "Sent"
=> Account settings / Copies & Folders / "When a message is sent, aromatically put a copy in ..." (I'm translating from French layout). This is what I do for each of my accounts in order to have received and sent email all in their Account's Archive folder.
The statement is a bit ambiguous, at least in French, when stated "Put a copy of sent email". At first I thought it would be a copy of the email already in the Sent folder. In fact it means a (the) copy of the sent email itself. Hum... is that English correct ?!
The feature built into Thunderbird affects where the reply you composed is stored. This add-on affects where the original message (the one you're replying *to*) is stored. Hopefully that clarifies things a bit.