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We present the median-stacked Lyman-$α$ surface brightness profile of 968 spectroscopically selected Lyman-$α$ emitting galaxies (LAEs) at redshifts $1.9<z<3.5$ in the early data of the Hobby-Eberly Telescope Dark Energy Experiment (HETDEX). The selected LAEs are high-confidence Lyman-$α$ detections with large signal-to-noise ratios observed with good seeing conditions (point-spread-function full-width-at-half-maximum $<1.4"$), excluding active galactic nuclei (AGN). The Lyman-$α$ luminosities of the LAEs are $10^{42.4}-10^{43}\, \mathrm{erg}\, \mathrm{s}^{-1}$. We detect faint emission in the median-stacked radial profiles at the level of $(3.6\pm 1.3)\times 10^{-20}\,\mathrm{erg}\,\mathrm{s}^{-1}\,\mathrm{cm}^{-2}\,\mathrm{arcsec}^{-2}$ from the surrounding Lyman-$α$ halos out to $r\simeq 160$ kpc (physical). The shape of the median-stacked radial profile is consistent at $r<80\,\mathrm{kpc}$ with that of much fainter LAEs at $3<z<4$ observed with the Multi Unit Spectroscopic Explorer (MUSE), indicating that the median-stacked Lyman-$α$ profiles have similar shapes at redshifts $2<z<4$ and across a factor of $10$ in Lyman-$α$ luminosity. While we agree with the results from the MUSE sample at $r<80\,\mathrm{kpc}$, we extend the profile over a factor of two in radius. At $r>80\,\mathrm{kpc}$, our profile is flatter than the MUSE model. The measured profile agrees at most radii with that of galaxies in the Byrohl et al. (2021) cosmological radiative transfer simulation at $z=3$. This suggests that the surface brightness of a Lyman-$α$ halo at $r\lesssim 100$ kpc is dominated by resonant scattering of Lyman-$α$ photons from star-forming regions in the central galaxy, whereas at $r > 100$ kpc it is dominated by photons from galaxies in surrounding dark matter halos.