We show the existence of transcendental entire functions $f: \mathbb{C} \rightarrow \mathbb{C}$ with Hausdorffdimension 1 Julia sets, such that every Fatou component of $f$ has infinite inner connectivity. We also show that there exist singleton complementary components of any Fatou component of $f$, answering a question of Rippon+Stallard. Our proof relies on a quasiconformal-surgery approach. This is joint work with Jack Burkart.