Place potatoes in a large pot, cover with cold salted water, and bring to a boil. Cook for 15–20 minutes, or until tender when pierced with a fork.
Drain the potatoes and return them to the pot. Add butter (or olive oil) and milk, then mash until smooth and creamy.
Stir in pesto until evenly combined. Taste and adjust seasoning with salt and pepper.
Spoon into a serving bowl and drizzle with a little extra pesto. Sprinkle with Parmesan.
