  The current consensus on the formation and evolution of the brightest cluster
galaxies is that their stellar mass forms early ($z \gtrsim 4$) in separate
galaxies that then eventually assemble the main structure at late times ($z
\lesssim 1$). However, advances in observational techniques have led to the
discovery of protoclusters out to $z \sim 7$, suggesting that the late-assembly
picture may not be fully complete. If these protoclusters assemble rapidly in
the early universe, they should form the brightest cluster galaxies much
earlier than suspected by the late-assembly picture. Using a combination of
observationally constrained hydrodynamical and dark-matter-only simulations, we
show that the stellar assembly time of a sub-set of the brightest cluster
galaxies occurs at high redshifts ($z > 3$) rather than at low redshifts ($z <
1$), as is commonly thought. We find, using isolated non-cosmological
hydrodynamical simulations, that highly overdense protoclusters assemble their
stellar mass into brightest cluster galaxies within $\sim 1$ $\mathrm{Gyr}$ of
evolution -- producing massive blue elliptical galaxies at high redshifts ($z
\gtrsim 1.5$). We argue that there is a downsizing effect on the cluster scale
wherein some of the brightest cluster galaxies in the cores of the most-massive
clusters assemble earlier than those in lower-mass clusters. In those clusters
with $z = 0$ virial mass $\geqslant 5\times10^{14}$ M$_\mathrm{\odot}$, we find
that $9.8$% have their cores assembly early, and a higher fraction of $16.4$%
in those clusters above $10^{15}$ M$_\mathrm{\odot}$. The James Webb Space
Telescope will be able to detect and confirm our prediction in the near future,
and we discuss the implications to constraining the value of
$\sigma_\mathrm{8}$.
