  Recent data from the \emph{Fermi} Large Area Telescope have revealed about a
dozen distant hard-spectrum blazars that have very-high-energy (VHE; $\gtrsim
100$ GeV) photons associated with them, but most of them have not yet been
detected by imaging atmospheric Cherenkov telescopes. Most of these high-energy
gamma-ray spectra, like those of other extreme high-frequency peaked BL Lac
objects, can be well explained either by gamma rays emitted at the source or by
cascades induced by ultra-high-energy cosmic rays, as we show specifically for
KUV 00311$-$1938. We consider the prospects for detection of the VHE sources by
the planned Cherenkov Telescope Array (CTA) and show how it can distinguish the
two scenarios by measuring the integrated flux above $\sim 500$ GeV (depending
on source redshift) for several luminous sources with $z \lesssim 1$ in the
sample. Strong evidence for the origin of ultra-high-energy cosmic rays could
be obtained from VHE observations with CTA. Depending on redshift, if the often
quoted redshift of KUV 00311-1938 ($z = 0.61$) is believed, preliminary
H.E.S.S. data favor cascades induced by ultra-high-energy cosmic rays. Accurate
redshift measurements of hard-spectrum blazars are essential for this study.
