Cheeze whiz!
No seriously. There’s this really rock hard Italian cheese we get. It tastes like parmesean, but it’s not parmesean. It’s really hard and salty. I bet that would pair nicely with an onion. And some of my Vieux-Chateau-du-Roi. 
After this thread yesterday I toasted a bagel with a tomato, onion slice, topped with store brand sharp cheddar and gorgonzola. Little mayo, salt and pepper. Yum.
Raw onions go best with real English farmhouse cheddar, like from Neal’s Yard – Keen’s cheddar or Montgomery’s cheddar or Borough Market cheddar or Quicke’s cheddar
Some cornichons … coarse ground mustard …
Get some fresh crusty bread with that – a baguette right out of the oven, steaming when you break it open …
I thought its culture was extinct and they couldn’t even find it in the back of a fridge in Australia. I’ll believe it when I taste it. Which I hope will be soon, because fake Liederkranz could be pretty good.
We are talking about raw onion and cheese sandwiches. Preferably on a rye that fights back. And maybe with some mustard. Who of us is NOT from the Upper Midwest of the USA?