Hi,
The software and driver downloads page for an HP Officejet Pro 8600 Plus e-All-in-One Printer - N911g has a reference to HP Easy Start for macOS 12.
https://support.hp.com/us-en/drivers/selfservice/hp-officejet-pro-8600-e-all-in-one-printer-series---n9/4322915/model/4323647
However, for earlier operating system versions, problems have been reported with HP Easy Start in some cases. HP Easy Admin has been a workaround. HP Easy Admin allows downloads without a connected printer. If necessary, this tool could perhaps be worth a try.
https://support.hp.com/us-en/document/c06164609
If nothing else works, an HP Officejet Pro 8600 (version 4.1.1) driver is mentioned in the article Printer and scanner drivers for Mac - Apple Support. The download from Apple can be found at HP 5.1.1 Printer Software Update. The system requirements say that macOS 12 is not supported, but reports indicate that the package has been used with some other printer models under said system anyway. It may or may not work with your printer, computer, and operating system version. Look upon it as an experiment. Make sure that you have backup copies of important files (documents, photos, et cetera) before any tests.
Unfortunately, an HP 8600 printer does not appear to be listed in (the April 15, 2022 edition of) About AirPrint - Apple Support. Three HP Officejet Pro 8600A models were shown in an older (February 13, 2012) HT4356 iOS: AirPrint 101 Apple article, though. Possibly an early firmware/AirPrint version?
You may want to check whether your printer has the latest possible firmware (some firmware updates can be found on HP's download page for the printer, if macOS 10.15 or earlier is selected).
Reset the printing system on your Mac to solve a problem – Apple Support (UK)
Add a printer to your printer list so you can use it on Mac – Apple Support (UK)
