  In models with extended Higgs sectors, it is possible that the Higgs boson
discovered at the LHC is not the lightest one. We show that in a realistic
model (the Type I 2-Higgs Doublet Model), when the sum of the masses of a light
scalar and a pseudoscalar ($h$ and $A$) is smaller than the $Z$ boson mass, the
Electroweak (EW) production of an $hA$ pair can dominate over QCD production by
orders of magnitude, a fact not previously highlighted. This is because in the
$gg$-initiated process, $hA$ production via a resonant $Z$ in the $s$-channel
is prohibited according to the Landau-Yang theorem, which is not the case for
the $q\bar{q}$-initiated process. We explore the parameter space of the model
to highlight regions giving such $hA$ solutions while being consistent with all
constraints from collider searches, $b$-physics and EW precision data. We also
single out a few benchmark points to discuss their salient features, including
the $hA$ search channels that can be exploited at Run II of the LHC.
