It must be multiple times. But you can only pillage 1 building after each battle, so you are limited by how many battles you can win.
I think it's after each conquest; the rules even define it as "if you are the only player to have Units in an opponent controlled Province."
Which also includes unopposed invasions and the turns after that.
You can basically pillage (1 time per occupied province) as long as the controlling player still has buildings in that Province.