Is there any book on $1$-dimensional complex analysis, where all is written in the language of sheaf theory? It's clear, that a lot of constructions can be formulated in simplier way using it. There are a lot of such books of n-dimensional complex analysis. And what about 1-dimensional?
Robert Gunning's "old" Princeton-Yellow-Series book "Intro to Riemann Surfaces" (the first in a sequence of several books he wrote about Riemann surfaces and related matters...) systematically uses sheaf theory (albeit not the derived-functor version, but Cech). In my opinion, it wonderfully illustrates how sheaf theory can be used, in a very tangible example.