Bring a large pot of water to a boil, add potatoes and cook for 12-15 minutes until they are tender and can easily be pierced with a fork.
In a seperate small pot, bring water to a boil, gently add eggs, and boil until cooked to your liking; 8 minutes for medium, 10 minutes for hard-boiled, or 12 minutes for extra hard-boiled.
While the potatoes and the eggs are cooking, combine the olive oil, basil, garlic, pine nuts, parmesean cheese, a pinch of salt and pepper in a blender, and blend until smooth. You may need to add more oil to get the desired consistency, and season with additional salt and pepper to taste.
Once the eggs have finished cooking, remove them from the pot and submerge them in a bowl of cold water to cool. Once cool to the touch, peel the eggs and then cut them into quarters.
Once the potatoes are cooked, strain them and then run them under cold water to help stop the cooking process.
Once potatoes have cooled slightly, cut them into bite-size pieces, halved or quartered is fine, and add them to a large bowl.
Once all of the potatoes have been cut, cover the potatoes with the pesto and then toss to combine. Set aside and allow to cool. (This salad can also be served warm if you prefer.)
Once you are ready to serve, top potatoes with eggs, and season with additional parmeseam cheese, salt and pepper to taste.
This salad can be eaten immediately or stored in the fridge for up to 5 days.