If things don't turn out exactly as planned when cooking an omelet, don't fret. Broken omelets make great scrambled eggs.
Preheat an 8-inch pan over medium-low heat for 2 minutes. Add oil and swirl to coat. Season eggs with salt and pepper and pour into pan.
Cook, occasionally tilting pan and lifting up set egg. Cook until omelet is just set, 3 to 4 minutes (be careful to not let the bottom brown).
Spread half the omelet with pesto and flip over other side. Continue to cook until pesto is warm, about 30 seconds more.