I think at the moment you can only do it with one of these dongles from the hori chat headset or something similar... or until Nintendo actually integrate chat into the Switch.
They are not cheap but SteelSeries Arctis 3 Bluetooth lets you to hardwire to the switch, and wirelessly to a phone or laptop at the same time....both audios at once. but you have to get the bluetooth version. I am on the verge of ordering them not not sure.
I don't know if that splitter cable will work but cant say for certain as have never tried it myself. I heard people just buy that Hori headset to get that dongle and just use it with any headphones they have.