  The $X(6900)$ resonance, very recently discovered in the double-$J/\psi$
channel at LHCb experiment, has spurred intensive interest in unravelling the
nature of the fully charmed tetraquark state. The aim of this paper is to
present a model-independent theoretical framework to study the inclusive
production of this novel species of exotic hadrons, the resonances composed of
four heavy quark (commonly referred to as $T_{4c}$), at large $p_T$ in hadron
collision experiments. Appealing to asymptotic freedom and the fact $m_c\gg
\Lambda_{\rm QCD}$, we propose that the nonpertubative yet universal
gluon-to-$T_{4c}$ fragmentation function, can be decomposed into the product of
the perturbatively calculable short-distance coefficient and the long-distance
NRQCD matrix elements. We compute the short-distance coefficient at
lowest-order in $\alpha_s$ and velocity expansion. Adopting the diquark ansatz
to roughly estimate those not-yet-known NRQCD matrix elements, together with
the standard QCD factorization theorem, we predict the differential production
rates for the $T_{4c/4b}(0^{++})$ and $T_{4c/4b}(2^{++})$ at large $p_T$ in
$pp$ collision, which eagerly awaits the confrontation with the future LHC
experiments.
