  Today, the majority of the cosmic baryons in the Universe are not observed
directly, leading to an issue of "missing baryons" at low redshift.
Cosmological hydrodynamical simulations have indicated that a significant
portion of them will be converted into the so-called Warm-Hot Intergalactic
Medium (WHIM), with gas temperature ranging between 10$^5$-10$^7$K. While the
cooler phase of this gas has been observed using O VI and Ne VIII absorbers at
UV wavelengths, the hotter fraction detection relies mostly on observations of
O VII and O VIII at X-ray wavelengths. Here, we target the forbidden line of
[Fe XXI] $\lambda$ 1354$\unicode{x212B}$ which traces 10$^7$K gas at UV
wavelengths, using more than one hundred high-spectral resolution
(R$\sim$49,000) and high signal to noise VLT/UVES quasar spectra, corresponding
to over 600 hrs of VLT time observations. A stack of these at the position of
known DLAs lead to a 5-$\sigma$ limit of $\mathrm{log[N([Fe\,XXI])]<}$17.4
(${EW_{rest}<22}$m$\unicode{x212B}$), three orders of magnitude higher than the
expected column density of the WHIM $\mathrm{log[N([Fe\,XXI])]<}$14.5. This
work proposes an alternative to X-ray detected 10$^7$K WHIM tracers, by
targeting faint lines at UV wavelengths from the ground benefiting from higher
instrumental throughput, enhanced spectral resolution, longer exposure times
and increased number of targets. The number of quasar spectra required to reach
this theoretical column density with future facilities including 4MOST,
ELT/HIRES, MSE and the Spectroscopic Telescope appears challenging at present.
Probing the missing baryons is essential to constrain the accretion and
feedback processes which are fundamental to galaxy formation.
