How to Make It
Cook the potatoes in a medium saucepan of boiling salted water until tender, about 15 minutes. Drain the potatoes and return them to the pan. Set the pan over moderately high heat and shake for about 1 minute to dry out the potatoes. Using a potato masher, mash the potatoes in the pan. Stir in the cream, Parmigiano-Reggiano and the 3 tablespoons of butter and season generously with salt and pepper. Scrape the mashed potatoes into a large bowl and set aside to cool to room temperature.
Preheat the oven to 475°. In a large stainless-steel bowl, beat the egg whites with a pinch of salt until soft peaks form. Stir one-third of the whites into the mashed potatoes to loosen them, then fold in the remaining whites with a rubber spatula until blended.
Butter 2 large rimmed baking sheets. Drop rounded tablespoons of the potato mixture onto the sheets about 1/2 inch apart. Flatten each mound slightly to form a disk. Brush the mounds with the melted butter and sprinkle with some grated cheese. Bake for about 10 minutes, or until lightly puffed and golden.
Meanwhile, in a bowl, combine the olives and basil. Top each puff with 1/2 teaspoon of the olive mixture and serve.
Make Ahead: The puffs can be prepared through Step 2 and refrigerated for up to 2 days. Or they can be baked earlier in the day and reheated in a 400° oven for 5 minutes.