Yes, you can "finish a project early" when its die is on 1.
During each week, the following things happen:
The active player draws a card, reads the relevant text aloud, and resolves it. They follow all bold text.
Project dice are reduced by 1, and any finished projects are updated.
The active player chooses and takes an action.
[...]
Unless your card specifically told you otherwise (in bold text), the next step is to reduce each project die on the map by 1. If a project reaches 0, the die gets removed and the project is completed. Whoever started the project gets to tell everyone how it turns out, and update the map to reflect its completion.
If a project finishes early (because a card says that "a project finishes early" in bold text), it is instead the responsibility of the active player to tell everyone how it turned out and update the map.
-- "The Week" and "Working on Projects", The Quiet Year
While you're reading the card the project hasn't finished. A card might also say, for instance, "choose one project, increase its project die by 3", and now the project that was on 1 won't finish this week but take three more weeks to come to completion.
It's the same with a project finishing early. The project finishes during the card phase, rather than the project phase, and the active player gets to determine how it turns out.
Now, there's very little reason to finish a project on 1 early; perhaps it's the only project and you'd rather not take the other option on the card, or perhaps the project is vital and the card says all other projects stall or increase their dice, and you'd rather not risk what happens next week. Absent these concerns, the practical upshot of choosing to do this is that you take control of someone else's project results away from them, since there is no doubt it would have finished anyway.
This is likely to get you some contempt. But maybe that's what you're gunning for? It's not unheard of for projects to be co-opted at the last minute by someone with a different agenda.
