The process of placing pages on a sheet is called Imposition. "Mutiple-fold" imposition of this kind is usually done only in commercial and large-scale printing, because the more pages on your sheet, the bigger the sheet needs to be. (An A3 sheet folded 3 times gives you a tiny A6 page. Your HP OfficeJet Pro 8600 only does A4!)
Large-scale printing devices may have options in their drivers to impose pages. There is also standalone Imposition software, which tends to manipulate PDFs as part of a pre-press workflow. That's probably your best bet for adding this feature.
There are plenty of cheap or free Imposition apps, which you can search for: though many of them will only do 4-page-per-sheet booklet, e.g. A4 page on A3 sheet. More complex imposition apps tend to be quite expensive (Imposition Wizard is $559, or $260 p.a.; Imposition Studio Digital on the Mac App Store is c. $250 -- and that's a 'Light' version of the $450 app!) because they are geared to commercial work.
Adobe's InDesign comes with Imposition built-in to its print options. There are also Adobe Acrobat imposition plug-ins.
If you are sending your PDFs to a commercial printer, then they will deal with the imposition that they need for their equipment.
It would be possible to write a script using MacOS CoreGraphics APIs (in python, AppleScriptObjC, or Swift) that would perform the necessary imposition on a PDF document.