We are interested in the structure of large Bienaymé-Galton-Watson random trees whose offspring distribution is critical and falls within the domain of attraction of a stable law of index $α=1$. In stark contrast to the case $α\in (1,2]$, we show that a condensation phenomenon occurs: in such trees, one vertex with macroscopic degree emerges. To this end, we establish limit theorems for centered downwards skip-free random walks whose steps are in the domain of attraction of a Cauchy distribution, when conditioned on a late entrance in the negative real line. These results are of independent interest. As an application, we study the geometry of the boundary of random planar maps in a specific regime (called non-generic of parameter $3/2$). This supports the conjecture that faces in Le Gall & Miermont's $3/2$-stable maps are self-avoiding.