Gnocchi with Pomodoro Sauce

Preheat oven to 350°F. Place potatoes directly on middle rack of preheated oven, and bake until tender, about 1 hour and 15 minutes. Remove from oven. Carefully cut potatoes in half lengthwise; let stand until cool enough to handle, about 20 minutes. Scoop potato flesh into a ricer; discard skins. Process potatoes through ricer into a large bowl. Add, but do not stir in, cheese, egg yolks, 1 1/2 teaspoons salt, and pepper. Sprinkle flour over mixture; using your hands, fold together until well incorporated and mixture forms a soft dough ball, about 1 minute. Wrap dough with plastic wrap; refrigerate 30 minutes. Remove dough from refrigerator, and unwrap. Place dough on a floured work surface, and cut dough evenly into 5 pieces. Using your hands, roll each piece into a 121/2-inch-long rope. Cut each rope into 1-inch pieces. Roll each piece until it is about 1 1/2 inches long. Place on a lightly floured baking sheet lined with parchment paper.

Bring 16 cups water and 1/4 cup salt to a boil in a large Dutch oven over high. Fill a large bowl with ice water. Add gnocchi to boiling water; cook, stirring occasionally, until gnocchi begin to float, about 2 minutes. Transfer to ice water; let cool 3 minutes. Drain well. Transfer gnocchi to a large bowl; add oil, and toss gently. Cover and refrigerate up to 3 hours. 

Process tomatoes in a blender until smooth, about 10 seconds. Transfer tomato puree to a large saucepan; add basil, garlic, and sugar. Bring to a simmer over medium-low. Simmer, stirring occasionally, until thickened, about 45 minutes. Add butter and remaining 1/2 teaspoon salt; stir until butter melts. 

Place gnocchi in tomato sauce; simmer over medium-low, stirring occasionally, until warmed, about 5 minutes. Garnish servings with additional cheese and basil, if desired.