Hangover Eggs are our family secret for the morning after a little too much alcohol! You can cook these even if you have a headache since all you need are a few simple ingredients and not much skill (!): eggs, olive oil, and feta cheese. Jokes aside, there are so many delicious Greek recipes for eggs, but perhaps none quite so satisfying as this one for Greek sunnyside up eggs with feta, aka Hangover Eggs. The secret is to cook them slowly and to spoon the hot Greek olive oil over the top of them while they fry.