You don't go head to head with them.
It is just like the question: how do I stop a monster truck with a grocery cart?
Answer: you don't.
League is not a 1v1 game.
I play Yi jungle. If I see a Rengar in the jungle with full ferocity, do I pop ult, E, and then Q onto him? lol no. I run the hell away and splitpush where he's not because he'll kill me in a single triple Q slash. Even if not, why risk it? There's no reason to fight Rengar. I splitpush better, I can take objectives, WHY FIGHT?
Evelynn is best used to feed your laners, then splitpush or tank in teamfights late game. You have a double on-hit applying nuke, use them to apply BoRK and Devourer/Wits End. Build Randuins, Tabis and Banshees to tank more. You have better chasing and are far harder to kite. Etc etc. Play to your strengths, 1v1ing the strongest 1v1 burst champs in the game is not one of them.