PuTTY Quick launch
If you find yourself in the same boat as me, which is that updated versions of PuTTy do not fix this issue, you can use the following workaround that I found here:
This will be an additional putty taskbar item if you already had the standard putty pinned to taskbar. Since I only use one putty session every time, I deleted the original putty taskbar pin. If you have multiple connections you want to pin, then you would repeat the above steps for each one and would probably want to change the shortcut icons for each to differentiate.