The tiny droplets of quark gluon plasma (QGP) created in high-energy nuclear collisions experience fast expansion and cooling with a lifetime of a few fm/c. Despite the information provided by probes such as jet quenching and quarkonium suppression, and the excellent description by hydrodynamical models, direct access to the time evolution of the system remains elusive. We point out that the study of hadronically decaying W bosons, notably in events with a top-antitop quark pair, can provide key novel insight into the time structure of the QGP. This is because of a unique feature, namely a time delay between the moment of the collision and that when the W-boson decay products start interacting with the medium. Furthermore, the length of the time delay can be constrained by selecting specific reconstructed top-quark momenta. We carry out a Monte Carlo feasibility study and find that the LHC has the potential to bring first limited information on the time structure of the QGP. Substantially increased LHC heavy-ion luminosities or future higher-energy colliders would open opportunities for more extensive studies.