You can't do this with a shortcut as there can be only one browser file association. You can, however, do what you want with a batch file(s).
Below is an example for you that will open a site in Internet Explorer. (My default browser is Firefox. I made this batch to enable quickly opening selected sites in IE.)
In this case I have a command line parameter so I can drag a link to it and I also added it as a Sent To shortcut. That enables right clicking on a saved link and sending to IE.cmd (the name of my batch file).
You could replace the %1 with a hard coded URL if you like for a specific site. Note of course, you will also need to set the program path and name per the browser you want to use.
start "" "\Program Files\Internet Explorer\iexplore.exe" %1
exit