I cannot categorically say that such fig does not exist that only produce one crop (breba) which grows on previous years growth where the leaves used to be, and does not grow any main crop figs that grow on the current year growth next to the leaves. But that is only my experience of not having grown/found such variety.
However, F4F website has a statement "Some varieties produce only a breba crop".
http://figs4fun.com/basics_Pruning.html
which says " Main crop figs form on the new growth that appears this season. Some varieties produce only a breba crop; some have both breba and main-crop figs, and many have only main-crop figs. If you prune a breba-only variety, you reduce or eliminate your fruit for that year. If you prune a variety with breba and main-crop figs, you sacrifice the breba crop. If you prune a main-crop-only variety, you lose nothing. In my opinion, breba-crop figs are not numerous, and mostly of inferior taste, sweetness and quality. MORE"