  The search for dark-matter (DM) candidates at high-energy colliders is one of
the most promising avenues to understand the nature of this elusive component
of the universe. Several searches at the Large Hadron Collider (LHC) have
strongly constrained a wide range of simplified models. The combination of the
bounds from the LHC with direct-detection experiments exclude the most minimal
scalar singlet DM model. To address this, Lepton portal DM models are suitable
candidates where DM is predominantly produced at lepton colliders since the DM
candidate only interacts with the lepton sector through a mediator that carries
a lepton number. In this work, we analyse the production of DM pairs in
association with a Higgs boson decaying into two bottom quarks at future muon
colliders in the framework of the minimal lepton portal DM model. It is found
that the usual cut-based analysis methods fail to probe heavy DM masses for
both the resolved (where the decay products of the Higgs boson can be resolved
as two well-separated small-$R$ jets) and the merged (where the Higgs boson is
clustered as one large-$R$ jet). We have then built a search strategy based on
Boosted-Decision Trees (BDTs). We have optimised the hyperparameters of the BDT
model to both have a high signal-to-background ratio and to avoid overtraining
effects. We have found very important enhancements of the signal significance
with respect to the cut-based analysis by factors of $8$--$50$ depending on the
regime (resolved or merged) and the benchmark points. Using this BDT model on a
one-dimensional parameter space scan we found that future muon colliders with
$\sqrt{s}=3$ TeV and ${\cal L} = 1~{\rm ab}^{-1}$ can exclude DM masses up to
$1$ TeV at the $95\%$ CL.
