We explore the phenomenology of models containing one Vector-Like Quark (VLQ), t , which can decay into the Standard Model (SM) top quark, t, and a new spin-0 neutral boson, S, the latter being either a scalar or pseudoscalar state. We parametrise the underlying interactions in terms of a simplified model which enables us to capture possible Beyond the SM (BSM) scenarios. We discuss in particular three such scenarios: one where the SM state is supplemented by an additional scalar, one which builds upon a 2-Higgs Doublet Model (2HDM) framework and another which realises a Composite Higgs Model (CHM) through partial compositeness. Such exotic decays of the t can be competitive with decays into SM particles, leading to new possible discovery channels at the Large Hadron Collider (LHC). Assuming t pair production via strong interactions, we design signal regions optimised for one t → S t transition (while being inclusive on the othert decay, and vice versa), followed by the decay of S into the two very clean experimental signatures S → γ γ and S → Z(→ + −)γ. We perform a dedicated signalto-background analysis in both channels, by using Monte Carlo (MC) event simulations modelling the dynamics from the proton-proton to the detector level. Under the assumption of BR(t → S t) = 100%, we are therefore able to realistically quantify the sensitivity of the LHC to both the t and S masses, assuming both current and foreseen luminosities. This approach paves the way for the LHC experiments to surpass current VLQ search strategies based solely on t decays into SM bosons (W ± , Z, h).