Heat the oven to 350° F degrees. Pierce the potatoes with a fork and bake until soft, or for about 90 minutes. While still hot, cut the potatoes in half and scoop out the middle. When cooled, pass the potato through a potato grater. Add the white pepper and then the egg yolk. Add about 1 cup of flour to the mixture. Mix thoroughly with a fork or pastry cutter and then form a ball. On a well-floured work surface, knead and fold mixture until the flour is incorporated, adding another ½ cup along the way. The dough should not feel sticky, so add a little more flour if necessary. Cover with flour and let rest for five minutes. Divide dough into 8 pieces. Roll out each piece into a long tube of about 1 inch thick and then cut into lengths about 1 inch long. Next, use a fork to make each piece into the shape of gnocchi, by rolling each piece up and down the tines of the fork. (If you do not plan to use the gnocchi within an hour or so, freeze them until ready.)
For the sauce:
Wash and dry the tomatoes. Take about a quarter of them and cut them in half. Set them aside for garnish. Place the remaining tomatoes in a blender or food processor. Pulse lightly until coarsely chopped, but not pureed. In a large skillet, sauté the garlic in the olive oil until golden. Remove the garlic with a slotted spoon. Add the tomatoes to the olive oil. Add half of the basil leaves and a teaspoon of each salt and pepper. Cook over medium heat for about 20 or so minutes, until slightly thickened.
Bring a pot of salted water to a boil. Carefully drop the gnocchi in water and cook until they float to the surface- 2-3 minutes. Carefully drain and add the gnocchi to the tomato mixture. Heat together for about a minute. Serve with a garnish of the remaining tomato halves and basil. Serve immediately with a sprinkle of Parmigiano.