  Within the context of composite Higgs models, recent hints on lepton-flavour
non-universality in $B$ decays can be explained by a vector resonance $V$ with
sizeable couplings to the Standard Model leptons ($\ell$). We argue that, in
such a case, spin-$1/2$ leptonic resonances ($L$) are most probably light
enough to open the decay mode $V \rightarrow L\ell$. This implies, in
combination with the fact that couplings between composite resonances are much
larger than those between composite and elementary fields, that this new decay
can be important. In this paper, we explore under which conditions it dominates
over other decay modes. Its discovery, however, requires a dedicated search
strategy. Employing jet substructure techniques, we analyse the final state
with largest branching ratio, namely $\mu^+\mu^- Z/h, Z/h \rightarrow$ jets. We
show that (i) parameter space regions that were believed excluded by di-muon
searches are still allowed, (ii) these regions can already be tested with the
dedicated search we propose and (iii) $V$ masses as large as $\sim 3.5$ TeV can
be probed at the LHC during the high-luminosity phase.
