Boil the potatoes with their skins on, drain when done, peel them, and puree them into a bowl using a food mill or a potato ricer.
Turn on the broiler.
Put two tablespoons of butter and chopped onion in a small skillet, turn on the heat to medium, and cook the onion, stirring from time to time, until it becomes colored a deep gold. Empty the contents of the pan over the potatoes in the bowl.
Lightly beat the eggs in a deep dish, then put them in the bowl with the potatoes. Add salt and pepper, and turn over the contents of the bowl with a fork and spoon to amalgamate them.
Add the basil and Parmesan and mix thoroughly once more.
Put 1tablespoon of butter in a 12-inch nonstick skillet and turn on the heat to medium. Do not let the butter become colored, but as soon as it begins to foam, pour the egg and potato mixture into the pan. Turn the heat down to very low. When the eggs have set and thickened, and they are creamy only in the center of the pan, run the skillet briefly under the broiler. Take it out the moment the surface of the frittata has set, but not browned.