  At the LHC, a TeV-scale leptoquark (LQ) that decays dominantly to a top quark
($t$) and a light charged lepton ($\ell=e,\mu$) would form a resonance system
of \emph{boosted-$t$ $+$ high-$p_{\rm T}$-$\ell$}. We consider all possible
vector LQ models within the Buchm\"{u}ller-R\"{u}ckl-Wyler classifications with
the desired decay. We propose simple phenomenological Lagrangians that are
suitable for bottom-up/experimental studies and, at the same time, can cover
the relevant parameter spaces of these models. In this simplified framework, we
study the pair and single production channels of vector LQs at the LHC.
Interestingly, we find that, like the pair production, the cross sections of
some single production processes also depend on the parameter $\kappa$ that
appears in the gluon-vector LQ coupling. We adopt a search strategy of
selecting events with at least one boosted hadronic top quark and exactly two
high-$p_{\rm T}$ leptons of the same flavor and opposite sign. This combines
events from the pair and single production processes and, therefore, can
enhance the discovery potential beyond that of the pair-production-only
searches. For $5\sigma$ discovery we find that vector LQs can be probed up to
$2.55$ TeV for $100\%$ branching ratio in the $t\ell $ decay mode and
$\mathcal{O}(1)$ new couplings at the $14$ TeV LHC with $3$ ab$^{-1}$ of
integrated luminosity.
