Spread the pesto in an even layer in an 8-inch nonstick skillet; bring to a gentle simmer over medium-low heat, 1 to 2 minutes. Add the eggs on top of the pesto and season with 1/4 teaspoon salt and a few grinds of pepper; continue to cook until the egg whites are just set and the oil has separated from the pesto, 3 to 4 minutes. Remove from the heat and cover with a tight-fitting lid (or a heatproof plate or a baking sheet). Let sit until the whites are completely set and the yolks are still runny, 1 to 2 minutes.