  We critically (re)examine the associated production process of a $J/\psi$
meson plus an open charm hadron at the LHC in the proton-proton ($pp$) and
proton-lead ($p{\rm Pb}$) collisions. Such a process is very intriguing in the
sense of tailoring to explore the double parton structure of nucleons and to
determine the geometry of partons in nuclei. In order to interpret the existing
$pp$ data with the LHCb detector at the center-of-mass energy $\sqrt{s}=7$ TeV,
we introduce two overlooked mechanisms for the double parton scattering (DPS)
and single parton scattering (SPS) processes. Besides the conventional DPS
mode, where the two mesons are produced almost independently in the two
separate scattering subprocesses, we propose a novel DPS mechanism that the two
constituent (heavy) quarks stemming from two hard scatterings can form into a
composite particle, like the $J/\psi$ meson, during the hadronization phase.
However, it turns out the corresponding contribution is small in
$J/\psi+c\bar{c}$ hadroproduction. On the contrary, we point out that the
resummation of the initial state logarithms due to gluon splitting into a charm
quark pair is crucial to understand the LHCb measurement, which was overlooked
in the literature. We perform a proper matching between the perturbative
calculations in different initial-quark flavor number schemes, generically
referring to the variable flavor number scheme. The new variable flavor number
scheme calculation for the process strongly enhance the SPS cross sections,
almost closing the discrepancies between theory and experiment. Finally, we
present our predictions for the forthcoming LHCb measurement in $p{\rm Pb}$
collisions at $\sqrt{s_{NN}}=8.16$ TeV. Some interesting observables are
exploited to set up the control regions of the DPS signal and to probe the
impact-parameter dependent parton densities in lead.

High Energy Physics - Phenomenology;High Energy Physics - Experiment;Nuclear Experiment;Nuclear Theory