  The origin of the excess of low-energy events observed by the MiniBooNE
experiment remains a mystery, despite exhaustive investigations of backgrounds
and a series of null measurements from complementary experiments. One
intriguing explanation is the production of beyond-the-Standard-Model particles
that could mimic the experimental signature of additional $\nu_e$ appearance
seen in MiniBooNE. In one proposed mechanism, muon neutrinos up-scatter to
produce a new ``dark neutrino'' state that decays by emitting highly-collimated
electron-positron pairs. We propose high-energy neutrinos produced from $W$
boson decays at the Large Hadron Collider as an ideal laboratory to study such
models. Simple searches for a low-mass, boosted di-lepton resonance produced in
association with a high-$p_\text{T}$ muon from the $W$ decay with Run 2 data
would already provide unique sensitivity to a range of dark neutrino scenarios,
with prompt and displaced searches providing complementarity. Looking farther
ahead, we show how the unprecedented sample of $W$ boson decays anticipated at
the HL-LHC, together with improved lepton acceptance would explore much of the
parameter space most compatible with the MiniBooNE excess.
