send notification when condition is met only on first instance

Maybe add a form rule to say

When the record is saved
and
Project is over budget (this would be some kind of formula checkbox field done as a lookup from the Parent Project record.

Change the manual checkbox field called  [Project is now over budget (manual checkbox)] to checked.

Then do not fire the Notification once that box is checked.

You will need to test if this will catch it in time.  It's possible that the email will not fire until the second expense over budget comes in.  I have not tested that.