You need what is called an "interrupted" projection, which (to the best of my knowledge) is not (yet) supported by
proj, the open-source projection engine used under the hood by QGIS.
However, all is not lost. Such projections are supported by extensions of
d3.js. See https://www.d3indepth.com/geographic/ for a tutorial on using
d3.js for map-making, and https://github.com/d3/d3-geo-projection#interrupted-projections for support of interrupted projections in particular.
Finally, d3.js supports the plotting of geoJson data, so that is how (I think, have not tried this) you can bridge the gap to QGIS, exporting (vector) layer data in this format.
A number of examples are shown at https://www.jasondavies.com/maps/. You'll have to inspect the source to see how it's been done; some are rendered pngs which won't be helpful, but some use the above techniques.