Using Chief Delphi as an example,
You can download the icon yourself and save it somewhere.
http://www.chiefdelphi.com/favicon.ico
Then you can go to your desktop, right click the shortcut, go the properties.
Somewhere in there should be a Change Icon button (in Vista it's under the Web Document tab). Click it, find where you saved the icon. As long as the .ico file stays where you saved it, that'll be the icon for that shortcut. Sanddrag is probably right and the file is being saved in the cache which is clearing every once and awhile.
You can do this with other sites too. The file is usually called 'favicon.ico' and is in the root directory of the site. (E.g.:
http://www.google.com/favicon.ico http://slashdot.org/favicon.ico etc) You can rename it when you save it (you should probably preserve the .ico extension)