I have not seen any games that do this, but I can think of a few ways to achieve this. Keep in mind that any solution will be browser and OS dependent, so you must offer alternative ways to start your game.
In general, my approach would consist of running your game's executable file from the browser. Here are some ideas I just made out. I'm not really sure whether or not they work for your purposes, or if they even work at all, but they may give you some ideas.
WScript.Shell. Only works on IE, but is probably your most flexible choice.
The mozilla.org/process XPCOM component would do the same thing in Firefox.
Execute the program from a signed Java applet. I haven't done applets in ages, but last time I checked, you could access local data XOR the internet from an applet, if it is properly signed.
Execute the program from a flash applet. This would be very similar to the Java applet.
Have the browser download a file with a custom extension, which your game previously registered. This requires a confirmation dialog, but is probably the most portable.
Have the link point to a custom URL scheme/activity which was previously registered by your game. This only works on iOS/Android, and is the mobile analog of the previous point.
Use a plugin. Most likely your game installer requires admin privileges, so use these privileges to install plugins for the currently found browser(s).
Make the button into a link to the locally installed game executable (via file://). This will fail when the game is not installed though.
But, if you can find a working example, you can check the page source, and at least get a hint of how they're doing it.
