Cook the eggs: Bring a pot of water to a boil. Cook the eggs for 8 minutes, then transfer them to cold water to stop the cooking process. Peel the eggs once cooled.
Prepare the pesto: Place basil leaves, olive oil, grated cheese (80 g), pecorino, pine nuts, and garlic clove in a container. Blend with an immersion blender until smooth.
Make the filling: Cut the peeled eggs in half and remove the yolks. In a bowl, mash the yolks with a fork and mix in the freshly made pesto.
Fill the egg whites: Fill a piping bag with the yolk-pesto mixture and pipe it into the hollowed egg whites.
Garnish and serve: Sprinkle the filled eggs with the remaining grated cheese (50 g) and serve immediately.