The Universe is experiencing an era of relative calm during "Cosmic Morning" at 4<z<5, in-between the intense formation of the first galaxies during Cosmic Dawn (and H reionisation) and the peak of star-formation density and quasar activity at Cosmic Noon (and He reionisation). Many global properties can only be studied with high-resolution quasar sightline spectroscopy during Cosmic Morning. Our program targets 30 bright quasars with GHOST, aiming to triple the existing data. The discovery by JWST of plentiful early Active Galactic Nuclei (AGN) has re-ignited debate on their contributions to H reionisation. Even a moderate AGN contribution would raise the temperature of the intergalactic medium (IGM) during Cosmic Morning, change the timing of both reionisations, and introduce telltale temperature fluctuations: all metrics to which our observations will bring improvements of factors 3-5. The smallest scales of the Lyman-alpha forest during Cosmic Morning also provide the strongest constraints on the mass of a potential Relic Warm Dark Matter (WDM) particle. Our sample will rule out (or detect!) the signature of a WDM particle with a mass up to 10 keV, when currently only masses <4 keV are strongly excluded. Closer to the quasar themselves, we will directly measure the temperature of the IGM inside quasar proximity zones, which are heated by He photo-dissociation. This measurement will be at least twice better than previous samples, and enable a search for trends in heating output with other quasar properties. Outside of the Lyman-alpha forest, we will uncover unprecedented numbers of weak, resolved metal absorber systems imprinted by the circumgalactic media (CGM) of foreground galaxies. We will fill in gaps in our knowledge of cosmic metal enrichment at 3.5<z<5 and at the weak end. Beyond the CGM, we expect the first detection of the CIV forest: an auto-correlation signal caused by contamination of the Cosmic Web itself by metals. Even a non-detection would pose the strongest constraint to date on the global volume-filling fraction of metals in the IGM. Despite tripling the number of high-resolution observations of quasars during Cosmic Morning, our program requires less than the total time invested until now at equal SNR per object. This is possible due to new quasars found by SkyMapper, which has now completed the search for bright quasars in the Southern Hemisphere to a completeness of 80%. Better targets will not be found even by Euclid and LSST, underlying the huge legacy value of these observations. Our team is committed to releasing to the community spectra reduced to the highest standard in order to advance our collective knowledge of astronomy across these far-reaching topics as well as many others.