  We present the first results from a recently concluded study of GRBs at
$z\gtrsim5$ with the Karl G. Jansky Very Large Array (VLA). Spanning $1$ to
$85.5$ GHz and 7 epochs from 1.5 to 82.3 d, our observations of GRB 140311A are
the most detailed joint radio and millimeter observations of a GRB afterglow at
$z\gtrsim5$ to date. In conjunction with optical/near-IR and X-ray data, the
observations can be understood in the framework of radiation from a single
blast wave shock with energy $E_{\rm K,iso}\approx8.5\times10^{53}$ erg
expanding into a constant density environment with density, $n_0\approx8\,{\rm
cm}^{-3}$. The X-ray and radio observations require a jet break at $t_{\rm
jet}\approx0.6$ d, yielding an opening angle of $\theta_{\rm
jet}\approx4^{\circ}$ and a beaming-corrected blast wave kinetic energy of
$E_{\rm K}\approx2.2\times10^{50}$ erg. The results from our radio follow-up
and multi-wavelength modeling lend credence to the hypothesis that detected
high-redshift GRBs may be more tightly beamed than events at lower redshift. We
do not find compelling evidence for reverse shock emission, which may be
related to fast cooling driven by the moderately high circumburst density.
