Preheat the oven to 400 degrees F.
Add the potatoes to a large pot of heavily salted water (about 1/4 cup salt) and bring to a boil. Reduce to a simmer and cook until a fork slides easily into a potato, about 15 minutes. Drain.
Generously oil a baking sheet with about 1/4 cup oil, making sure to cover the whole sheet. Arrange the potatoes on the baking sheet and gently smoosh them with a potato masher or cup -- not so much that it breaks the bottom skin or crumbles the potatoes, just enough to break open the top.
Drizzle each potato with olive oil or a tablespoon of melted butter, if you prefer. Season generously with salt and pepper.
Bake until golden brown, about 30 minutes.
Serve with roasted garlic aioli or a bit of sour cream and fresh herbs.
Combine the egg yolks, roasted garlic, lemon juice, mustard, paprika and salt in a blender. Blend on high to emulsify. With the motor running, slowly drizzle in the olive oil; the mixture will lighten and become thick and smooth. Taste and adjust the seasoning as needed.
Serve right away, or refrigerate for up to 1 day.
Consumption of raw or undercooked eggs, shellfish and meat may increase the risk of foodborne illness.