  Reionization-era galaxies tend to exhibit weak Ly$\alpha$ emission, likely
reflecting attenuation from an increasingly neutral IGM. Recent observations
have begun to reveal exceptions to this picture, with strong Ly$\alpha$
emission now known in four of the most massive z=7$-$9 galaxies in the CANDELS
fields, all of which also exhibit intense [OIII]$+$H$\beta$ emission (EW$>$800
$\mathrm{\mathring{A}}$). To better understand why Ly$\alpha$ is anonymously
strong in a subset of massive z$\simeq$7$-$9 galaxies, we have initiated an
MMT/Binospec survey targeting a larger sample (N=22) of similarly luminous
($\simeq$1$-$6 L$^{\ast}_{\mathrm{UV}}$) z$\simeq$7 galaxies selected over very
wide-area fields ($\sim$3 deg$^2$). We confidently ($>$7$\sigma$) detect
Ly$\alpha$ in 78% (7/9) of galaxies with strong [OIII]$+$H$\beta$ emission
(EW$>$800 $\mathrm{\mathring{A}}$) as opposed to only 8% (1/12) of galaxies
with more moderate (EW=200$-$800 $\mathrm{\mathring{A}}$) [OIII]$+$H$\beta$. We
argue that the higher Ly$\alpha$ EWs of the strong [OIII]$+$H$\beta$ population
likely reflect enhanced ionizing photon production efficiency owing to their
large sSFRs ($\gtrsim$30 Gyr$^{-1}$). We also find evidence that Ly$\alpha$
transmission from massive galaxies declines less rapidly over $6<z<7$ than in
low-mass lensed systems. In particular, our data suggest no strong evolution in
Ly$\alpha$ transmission, consistent with a picture wherein massive z$\simeq$7
galaxies often reside in large ionized regions. We detect three
closely-separated ($R$ = 1.7 physical Mpc) z$\simeq$7 Ly$\alpha$ emitters in
our sample, conceivably tracing a large ionized structure that is consistent
with this picture. We detect tentative evidence for an overdensity in this
region, implying a large ionizing photon budget in the surrounding volume.
