Heat oil in 12-inch nonstick skillet over low heat for 5 minutes. Meanwhile crack 2 eggs into a small bowl and season with salt and pepper. Repeat with remaining 2 eggs in a second bowl.
Increase heat to medium-high and heat until oil is shimmering. Add butter to skillet and quickly swirl to coat pan. Working quickly, pour 1 bowl eggs into one side of pan, and 1 bowl on other side. Cover and cook for 1 minute.
Remove skillet from burner and let stand, covered, 15-45 seconds for runny yolks, 45-60 seconds for soft but set yolks, and about 2 minutes for medium set yolks. Slide eggs onto plate and serve.
