Place the potatoes, garlic, and a tablespoon of salt in a large pot and fill with water.
Bring to a boil over high heat and then reduce the heat to medium, simmer 10-15 minutes or until the potatoes are just fork tender and drain.
Meanwhile, make the pesto in a blender or food processor, combine the cooked garlic, basil, pine nuts, parmesan, olive oil, lemon zest, lemon juice, and crushed red pepper flakes. Pulse until chunky smooth.
Cut the potatoes in half and add to a large serving bowl, tossing them with the butter.
Reserve 1/4 cup pesto for serving, and then add the remaining pesto and a pinch of pepper to the bowl with the potatoes.
Toss well to combine.
Taste and adjust salt and pepper to your liking.
Serve warm or cover and place in the fridge.
Add the eggs just before serving and drizzle with reserved 1/4 cup pesto.