Crack the eggs into a large bowl and whisk together until light and airy.
Melt the butter in a medium skillet over low heat. Add the eggs to the pan and wait about 10 seconds for them to start to set up. Swirl them with a rubber spatula to create soft curds. Season with salt and continue swirling until the curds are set but still a little wet.
Garnish with chives, and drizzle with extra-virgin olive oil.