In a liquid measuring cup, add 1 Tablespoon of apple cider vinegar. Top with non-dairy milk to the one cup mark. This will make one cup of a buttermilk substitute. Let sit for 10 minutes.
While milk is curdling, slice onion very thin with mandoline. ALWAYS use the handguard with a mandoline, protect your hands!
Place the thinly sliced onion in a baking dish. Top with buttermilk. Let sit for at least one hour on the counter. If you prefer, you can even let them sit in the buttermilk overnight in the refrigerator.
Mix flour, black pepper and cayenne together in a small dish.
Heat oil that is 2" to 3" deep to 350 degrees.
Remove a handful of the onions from the milk. Toss in the flour mixture. Carefully drop into the hot oil.
Fry for 1 to 2 minutes, until lightly browned. I give mine a quick stir about halfway through to keep them from clumping.
Remove from oil with slotted spoon and place on paper towel lined baking sheet.
If making ahead of time, store in an airtight container until used. I have stored these for up to 3 days before using.