  We explore the possibility of planet formation in the carbon-rich
protoplanetary disks of carbon-enhanced metal-poor (CEMP) stars, possible
relics of the early Universe. The chemically anomalous abundance patterns
([C/Fe] $\geq$ 0.7) in this subset of low-mass stars suggest pollution by
primordial core-collapsing supernovae (SNe) ejecta that are particularly rich
in carbon dust grains. By comparing the dust-settling timescale in the
protoplanetary disks of CEMP stars to the expected disk lifetime (assuming
dissipation via photoevaporation), we determine the maximum distance $r_{max}$
from the host CEMP star at which carbon-rich planetesimal formation is
possible, as a function of the host star's [C/H] abundance. We then use our
linear relation between $r_{max}$ and [C/H], along with the theoretical
mass-radius relation derived for a solid, pure carbon planet, to characterize
potential planetary transits across host CEMP stars. Given that the related
transits are detectable with current and upcoming space-based transit surveys,
we suggest initiating an observational program to search for carbon planets
around CEMP stars in hopes of shedding light on the question of how early
planetary systems may have formed after the Big Bang.
