We initiate the study of Iwasawa theory for branched $\mathbb{Z}_{p}$-towers of finite connected graphs. These towers are more general than what have been studied so far, since the morphisms of graphs involved are branched covers, a particular kind of harmonic morphisms of graphs. We prove an analogue of Iwasawa's asymptotic class number formula for the $p$-part of the number of spanning trees in this setting. Moreover, we find an explicit generator for the characteristic ideal of the torsion Iwasawa module governing the growth of the $p$-part of the number of spanning trees in such towers.