Had a look at the application itself, you can actually tell it to start minimized by checking "Start JoyToKey in a minimized mode" in the "Preferences" window of JoyToKey, if that's what you're after.
EDIT: Hah. If you've tried that already, I can see why you'd seek an alternative: JoyToKey won't actually come out of its minimized state when started this way, and has to be closed via task manager, after which the setting has to be undone by manually editing the config file
So, when you create a desktop shortcut to JoyToKey and select the, 'Minimized' option for Run, JoyToKey minimizes just fine and VoiceAttack's minimize does not work?
Indeed. Though, unlike with notepad, the title bar for JoyToKey appears when the application starts, and goes through a minimizing animation.