  A measurement of the $hh+2j$ channel at the LHC would be particularly
thrilling for electroweak physics. It is not only the leading process which is
sensitive to the $W^+W^- hh$ and $ZZhh$ interactions, but also provides a
potentially clear window to study the electroweak symmetry-breaking sector by
probing Higgs-Goldstone interactions through the weak boson fusion component of
the scattering process. Until now, a phenomenologically complete analysis of
this channel has been missing. This is mostly due to the high complexity of the
involved one-loop gluon fusion contribution and the fact that a reliable
estimate thereof cannot be obtained through simplified calculations in the
$m_t\to \infty$ limit. In particular, the extraction of the Higgs trilinear
coupling from this final state rests on a delicate $m_t$-dependent interference
pattern which is not captured in an effective field theory approach. In this
paper, we report on the LHC's potential to access di-Higgs production in
association with two jets in a fully-showered hadron-level analysis. Our study
includes the finite top and bottom mass dependencies for the gluon fusion
contribution. On the basis of these results, we also comment on the potential
sensitivity to the trilinear Higgs and $VV^\dagger hh$ ($V=W^\pm,Z$) couplings
that can be expected from measurements of this final state.
