  We present a detailed study concerning a new physics scenario involving four
fermion operators of the Nambu-Jona-Lasinio type characterized by a
strong-coupling ultraviolet fixed point where composite particles are formed as
bound states of elementary fermions at the scale $\Lambda ={\cal
O}(\text{TeV})$. After implementing the model in the Universal FeynRules Output
format, we focus on the phenomenology of the scalar leptoquarks at the LHC and
the High-Luminosity option. Leptoquark particles have undergone extensive
scrutiny in the literature and experimental searches, primarily relying on pair
production and, more recently, incorporating single, t-channel, and
lepton-induced processes. This study marks, for the first time, the examination
of these production modes at varying jet multiplicities. Novel mechanisms
emerge, enhancing the total production cross-section, especially for
leptoquarks couplings to higher fermion generations. A global strategy is
devised to capture all final state particles produced in association with
leptoquarks or originating from their decay, which we termed ``exclusive'', in
an analogy to the nomenclature used in nuclear reactions. The assessment of the
significance in current and future LHC runs, focusing on the case of leptoquark
coupling to a muon - $\textit{c}$ quark pair, reveals superior sensitivity
compared to ongoing searches. Given this heightened discovery potential, we
advocate the incorporation of exclusive leptoquark searches in future
investigations at the LHC.
