The Spektrum-Roentgen-Gamma (SRG)/extended Roentgen Survey with an Imaging Telescope Array (eROSITA) All-Sky Survey (eRASS) is expected to contain $ quasars that emitted their light when the universe was less than a billion years old, that is, at $z>5.6$. By selection, these quasars populate the bright end of the active galactic nuclei (AGN) X-ray luminosity function, and their space density offers a powerful demographic diagnostic of the parent super-massive black hole (SMBH) population. Of the $ 400$ quasars that have been discovered at $z>5.6$ to date, less than 15<!PCT!> have been X-ray detected. We present a pilot survey to uncover the elusive X-ray luminous end of the distant quasar population. We have designed a quasar selection pipeline based on optical, infrared and X-ray imaging data from DES DR2, VHS DR5, CatWISE2020 and the eRASS (up to its four-pass cumulative version, eRASS:4). The core selection method relies on SED template fitting. We performed optical follow-up spectroscopy with the Magellan /LDSS3 instrument for the redshift confirmation of a subset of candidates. We have further obtained a deeper X-ray image of one of our candidates with Chandra ACIS-S. We report the discovery of five new quasars in the redshift range $5.6 < z < 6.1$. Two of these quasars are detected in eRASS and are, therefore, X-ray ultra-luminous by selection. We also report the detection of these quasars at radio frequencies. The first one is a broad absorption line quasar, which shows significant, order-of-magnitude X-ray dimming over 3.5 years, corresponding to six months in the quasar rest frame. The second X-ray detected quasar is a jetted source with compact morphology. We show that a blazar configuration is likely for this source, making it one of the most distant blazars known to date. With our pilot study, we demonstrate the power of eROSITA as a discovery machine for luminous quasars in the epoch of reionization. The X-ray emission of the two eROSITA detected quasars are likely to be driven by different high-energetic emission mechanisms, a diversity which we will further explore in a future systematic full-hemisphere survey.