Measurement of double-charmonium production in $e^+e^-$ annihilation around $\sqrt{s} = 10.58$~GeV at {\sc BaBar}

Prompt production of $J/\psi$ or $\psi(2S)$ in association with a second charmonium state has been observed by both the BaBar and Belle experiments in $e^+e^-$ annihilation around $\sqrt{s} = 10.58$~GeV. These processes provide an opportunity to study both perturbative and non-perturbative effects in QCD and to search for new charmonium states recoiling against the reconstructed $J/\psi$ or $\psi(2S)$. I will present a study of such events using the full BaBar dataset, where the $J/\psi$ and $\psi(2S)$ are reconstructed via the decays $J/\psi \to \ell^+\ell^-$, $\psi(2S) \to J/\psi \pi^+\pi^-$, and $\psi(2S)\to \ell^+\ell^-$.