Shortcut Laakso spaces, pure PI unrectifiability and differentiability of Lipschitz functions

We construct a family of purely PI unrectifiable Lipschitz differentiability spaces and investigate the possible of Banach spaces targets for which Lipschitz differentiability holds. We provide a general investigation into the geometry of \emph{shortcut} metric spaces and characterise when such spaces are PI rectifiable, and when they are $Y$-LDS, for a given $Y$. The family of spaces arises as an example of our characterisations. Indeed, we show that Laakso spaces satisfy the required hypotheses.