Pasta Dough

1. Make a pile of flour on a work surface and form a well; add eggs and scramble with a fork, gradually incorporating flour until a shaggy dough forms.
2. Knead by hand until smooth, cover and rest 10–15 minutes, then knead until very smooth and elastic, about 4 more minutes; divide dough in half and refrigerate one piece at least 2 hours (freeze the other).
3. Shape one piece into a rectangle about 1 inch thick and roll out on a floured surface into a very thin large rectangle; turn and dust with flour as needed and cut lengthwise into 2 long strips.
4. Bring a large pot of lightly salted water to a boil; lower pasta strips and cook 1 minute 30 seconds, transfer to cold water until cooled, then cut each strip in half to make 4 pieces.

Cheese Filling

1. Combine ricotta, Parmesan, egg yolk, parsley, salt, black pepper, and cayenne in a bowl and stir until evenly combined; cover and refrigerate until needed.

Lasagna

1. Preheat oven to 450°F (230°C).
2. Set aside 1/2 cup of the cheese mixture; dollop remaining cheese evenly over the 4 pasta pieces and spread roughly, then spoon about 1/3 cup sauce onto each piece.
3. Top each piece with torn basil and scatter most of the shredded mozzarella, reserving 3/4 cup; fold each piece in half to form stuffed handkerchiefs.
4. Spread sauce on the bottom of a baking dish and place the first handkerchief; spread over 1/3 of reserved cheese, add a few spoons of sauce and some mozzarella; repeat with remaining handkerchiefs.
5. Top with remaining sauce, reserved mozzarella, and a dusting of Parmesan; drizzle with olive oil.
6. Bake until top is lightly golden and edges are bubbling, 25–30 minutes; let rest 10–15 minutes, then cut into quarters to serve.