Bring a large pot of salted water to a rolling boil and fill a bowl with ice water. Add the basil leaves to the boiling water and cook for exactly 15 seconds. Remove the basil with a slotted spoon and immediately plunge the leaves into the ice water to set the bright green color. Drain and set aside.
Peel the potatoes and cut them in quarters. Add the potatoes to the same pot of boiling water and return to a boil. Cook the potatoes for 20 to 25 minutes, until very tender. Drain well, return to the saucepan, and steam over low heat until any remaining water evaporates.
In a small saucepan over medium heat, heat the half-and-half and Parmesan cheese until the cream simmers. Place the basil in a food processor fitted with the steel blade and puree. Add the hot cream mixture and process until smooth.
With a handheld mixer with the beater attachment, beat the hot potatoes in the pot until they are broken up. Slowly add the hot basil cream, the salt, and pepper and beat until smooth. If the potatoes need to be reheated, cover and cook gently over low heat for a few minutes. Pour into a serving bowl, sprinkle with extra Parmesan cheese, season to taste, and serve hot.