1. Go to your Steam directory and find Steam.EXE in "C:\Program Files (x86)\Steam".
2. Right click Steam.EXE and go to Properties.
3. Go to compatibility tab and make sure all compatibility modes, including in all user settings, are UNCHECKED.
4. On Steam itself, add a non-steam game and just add Dolphin.EXE from your folder.
5. Now that it's added to steam, connect to your PC in the Shield Menu then select Steam.
6. Now go to your games lists and find "dolphin" and start it up. It'll appear on both screens. Just click a game and walla, you're playing a Gamecube on a portable.
Notes:
1. Before connecting your PC to the shield, make sure your PC has completely booted up if you've restarted or else you'll connect before some Nvidia processes are ready.
2. If it doesn't work, reboot your computer.
3. Add the steam directory in Geforce Experience if Steam doesn't show up