Can I Pin a Specific PDF to my Start Menu?

You can right-click any PDF file that you choose and select Send to > Desktop (create shortcut). You may also prefer to rename your shortcut to something more unique, or at the very least, remove the visible - Shortcut suffix.

After that, you can copy that shortcut to one of the following locations:

Only your Start Menu - %AppData%\Microsoft\Windows\Start Menu\Programs

All users Start Menu - %ProgramData%\Microsoft\Windows\Start Menu\Programs

Taking it a step further, you can potentially right-click that shortcut within the alphabetical All Apps list and select Pin to Start.