  Many scenarios of physics beyond the Standard Model that address the
hierarchy problem also predict the existence of vector-like top quark partners,
which are generally expected around the TeV scale. In this paper, we propose to
search for a vector-like top quark partner with charge $2/3$ in a simplified
model including only two free parameters, the coupling constant $g^{\ast}$ and
top quark partner mass $m_T$. We investigate the observability of the top quark
partner through the process $pp \to T(\to th)j\to t(\to b W^+\to b \ell^{+}
\nu_{\ell})h( \to WW^{\ast}\to \ell^{+}\nu\ell^{-}\bar{\nu}) j$, where $T$ is
the heavy top quark partner and $h$ the SM-like Higgs boson, at the Large
Hadron Collider (LHC). The discovery prospects and exclusion limits on the
parameter plane defined by ($m_T,g^{\ast}$) are obtained for the already
scheduled LHC runs as well as at the future High-Luminosity~LHC (HL-LHC). The
constraints and projected sensitivities are also interpreted in a realistic
model, i.e., the minimal Composite Higgs Model with singlet top quark partners.
We finally also analyze the projected sensitivity in terms of the production
cross section times branching fraction at the (HL-)LHC.
