  We present the discovery of the first millimeter afterglow of a
short-duration $\gamma$-ray burst (SGRB) and the first confirmed afterglow of
an SGRB localized by the GUANO system on Swift. Our Atacama Large
Millimeter/Sub-millimeter Array (ALMA) detection of SGRB 211106A establishes an
origin in a faint host galaxy detected in Hubble Space Telescope (HST) imaging
at $0.7\lesssim z\lesssim1.4$. From the lack of a detectable optical afterglow,
coupled with the bright millimeter counterpart, we infer a high extinction,
$A_{\rm V}\gtrsim2.6$ mag along the line of sight, making this the one of the
most highly dust-extincted SGRBs known to date. The millimeter-band light curve
captures the passage of the synchrotron peak from the afterglow forward shock
and reveals a jet break at $t_{\rm jet}=29.2^{+4.5}_{-4.0}$~days. For a
presumed redshift of $z=1$, we infer an opening angle, $\theta_{\rm
jet}=(15.5\pm1.4)$~degrees, and beaming-corrected kinetic energy of
$\log(E_{\rm K}/{\rm erg})=51.8\pm0.3$, making this one of the widest and most
energetic SGRB jets known to date. Combining all published millimeter-band
upper limits in conjunction with the energetics for a large sample of SGRBs, we
find that energetic outflows in high density environments are more likely to
have detectable millimeter counterparts. Concerted afterglow searches with ALMA
should yield detection fractions of 24-40% on timescales of $\gtrsim2$~days at
rates $\approx0.8$-1.6 per year, outpacing the historical discovery rate of
SGRB centimeter-band afterglows.
