Heat the oil in a medium nonstick skillet over low heat until slightly shimmering, about 5 minutes. Crack an egg into a small ramekin and slowly add it to the skillet; repeat with the other egg, adding it to the other side of the skillet. Cover with a tight lid and cook, uninterrupted, until the whites are completely set but the yolks are still runny, 2 to 2 1/2 minutes. Slide the eggs out of the skillet onto a plate or toast. Season with salt and pepper.