Try to think of what the Applet actually *is*: An online application embedded in a web page, nothing more, nothing less.
You don't want to restart the Applet, you want to restart your *game*, not the whole applet. Try to refactor your code, so you can simply call Game.start() or something and that your class extending Applet will be dumb enough to just show what your 'Game' class does without knowing much else.
A (shaky) analogy: When you start an application on your PC, you're not restarting your whole PC, are you?
(meant as constructive criticism