Texas Spectroscopic Search for Ly$\alpha$ Emission at the End of
  Reionization II. The Deepest Near-Infrared Spectroscopic Observation at
  $z\gtrsim7$
  Realizing the utility of Ly$\alpha$ emission to trace the evolution of the
intergalactic medium (IGM) during the epoch of reionization requires deep
spectroscopy across the boundary of optical and near-infrared (NIR)
spectrographs at $z\sim7.2$ when Ly$\alpha$ emission is at $\sim$1$\mu$m. Our
Texas Spectroscopic Search for Ly$\alpha$ Emission at the End of Reionization
includes 18 nights of deep spectroscopic observations using the Keck DEIMOS
(optical) and MOSFIRE (NIR) spectrographs. Within this dataset we observe
Ly$\alpha$ emission from 183 photometric-redshift selected galaxies at $z =$
5.5 - 8.3 from the Cosmic Assembly Near infrared Deep Extragalactic Legacy
Survey (CANDELS). Our overlapping MOSFIRE observations, over 84 galaxies total,
provide the deepest NIR spectroscopic data yet obtained for Ly$\alpha$ from
galaxies $z > 7$, with $>16$ hr integration time for four observed galaxies.
Here we analyze these four targets, and we report the discovery of a new $z =
7.60$ Ly$\alpha$ detection as well as provide an updated observation of the
previously confirmed $z=7.51$ Ly$\alpha$ emission from Finkelstein et al.
(2013) with a $\sim$3$\times$ longer exposure time. Our analysis of these
Ly$\alpha$ emission line profiles reveal a significant asymmetric shape. The
two detected Ly$\alpha$ emission lines from bright sources
($M_{\text{UV}}<-20.25$) could imply that these bright galaxies inhabit ionized
bubbles in a partially neutral IGM, although deeper exposures may yet reveal
Ly$\alpha$ emission in the fainter sources.
