Abstract: The $X(6900)$ resonance discovered in the double-$J/\psi$ channel at LHCb experiment has spurred intensive interest in unravelling the nature of the fully charmed tetraquark state. We present a model-independent theoretical framework to study the production of this novel species of exotic hadrons, both at large $p_T$ in hadron collision experiments and at B factory. Appealing to asymptotic freedom and the fact $m_c\gg \Lambda_{\rm QCD}$, we propose that quantities related to the production of $T_{4c}$, including the cross sections and the 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 coefficients at lowest-order in $\alpha_s$ and velocity expansion. Adopting three phenomenological models 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 $T_{4c}(0^{++})$ and $T_{4c}(2^{++})$ at large $p_T$ in $pp$ collision, which eagerly awaits the confrontation with the future LHC experiments. Such method is also applied to investigate the cross sections and angular distributions of $T_{4c}$ production at B factory.