Break the three eggs into a bowl and whisk them together with some pepper until well combined. Toast the pine nuts briefly in a dry skillet over medium-high heat until fragrant and light brown. Remove the nuts from the pan and set aside temporarily.
Heat the teaspoon of olive oil in the now-empty skillet. Add the fresh spinach and stir briefly until the leaves have just wilted. Pour the whisked eggs evenly over the warm spinach. Let the omelet cook gently over low heat until the bottom is cooked and the top is completely set.
Distribute the tablespoon of green pesto in small dots over the top of the cooked omelet. Slide a spatula under the eggs and fold the omelet carefully in half in the pan, so the heat softens the pesto slightly.
Slide the folded omelet directly from the skillet onto a flat plate. Sprinkle the toasted pine nuts evenly over it and serve the dish immediately.