We study the fragmentation of a jet propagating in a dense quark-gluon plasma. Using a leading, double-logarithmic approximation in perturbative QCD, we compute for the first time the effects of the medium on the vacuum-like emissions. We show that, due to the scatterings off the plasma, the in-medium parton showers differ from the vacuum ones in two crucial aspects: their phase-space is reduced and the first emission outside the medium can violate angular ordering. We compute the jet fragmentation function and find results in qualitative agreement with measurements at the LHC.