  The Future Circular Collider (FCC-ee) offers the unique opportunity of
studying the Higgs coupling to the electron, $y_e$, via resonant s-channel
production, $e^+e^- \to H$, in a dedicated run at $\sqrt{s} = m_H$. The
signature for direct Higgs production is a small rise in cross sections for
particular final states, consistent with Higgs decays, over the expectations
for their occurrence due to SM background processes involving $Z^*,\gamma^*$,
or t-channel exchanges. Performing such a measurement is remarkably challenging
for four main reasons. First, the low value of the e$^\pm$ mass leads to a tiny
$y_e$ coupling, and correspondingly small cross section: $\sigma_{ee\to
H}\,\propto m_e^2 = 0.57$ fb accounting for initial-state radiation. Second,
the $e^+e^-$ beams must be monochromatized such that their c.m. energy spread
is commensurate with the narrow width of the SM Higgs boson, $\Gamma_H = 4.1$
MeV, while keeping large beam luminosities. Third, the Higgs mass must also be
known beforehand with a few-MeV accuracy in order to operate the collider at
the resonance peak, $\sqrt{s} = m_H$. Last but not least, the cross sections of
the background processes are many orders-of-magnitude larger than those of the
Higgs decay signals. A generator-level study of 11 Higgs decays using a
multivariate analysis, exploiting BDTs to discriminate signal and background
events, identifies two final states as the most promising ones in terms of
statistical significance: $H\to gg$ and $H\to WW^*\to\ell\nu$ + 2 jets. For a
benchmark 4.1-MeV c.m. energy spread (leading to $\sigma_{ee\to H}\, = 0.28$
fb) and $\mathcal{L}_{int}=10$ ab$^{-1}$, a $1.3\sigma$ signal significance can
be reached, corresponding to an upper limit on the e$^\pm$ Yukawa at 1.6 times
the SM value: $|y_e|<1.6|y^{SM}_e|$ at 95\% confidence level, per IP per year.
Directions for future improvements of the study are outlined.
