  We present results from \textsc{GigaEris}, a cosmological, $N$-body
hydrodynamical "zoom-in" simulation of the formation of a Milky Way-sized
galaxy halo with unprecedented resolution, encompassing of order a billion
particles within the refined region. The simulation employs a modern
implementation of smoothed-particle hydrodynamics, including metal-line cooling
and metal and thermal diffusion. We focus on the early assembly of the galaxy,
down to redshift $z=4.4$. The simulated galaxy has properties consistent with
extrapolations of the main sequence of star-forming galaxies to higher
redshifts and levels off to a star formation rate of $\sim$60$\,
M_{\odot}$~yr$^{-1}$ at $z=4.4$. A compact, thin rotating stellar disk with
properties analogous to those of low-redshift systems arises already at $z \sim
8$. The galaxy rapidly develops a multi-component structure, and the disk, at
least at these early stages, does not grow "upside-down" as often reported in
the literature. Rather, at any given time, newly born stars contribute to
sustain a thin disk. The kinematics reflect the early, ubiquitous presence of a
thin disk, as a stellar disk component with $v_\phi/\sigma_R$ larger than unity
is already present at $z \sim 9$--10. Our results suggest that high-resolution
spectro-photometric observations of very high-redshift galaxies should find
thin rotating disks, consistent with the recent discovery of cold rotating gas
disks by ALMA. Finally, we present synthetic images for the JWST NIRCam camera,
showing how the early disk would be easily detectable already at those early
times.
