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. When too thick for a fork, use a bench scraper to bring in remaining flour and cut until dough comes together; knead by hand about 3 minutes until a smooth ball.
3. Cover and let rest 10–15 minutes, then knead about 4 minutes more until very smooth and elastic; divide dough in half, wrap, refrigerate one piece at least 2 hours (or up to overnight) and freeze the other.
4. Shape one piece into a rectangle ~1 inch thick and roll out on a lightly floured surface into a very thin rectangle, turning and dusting with flour as needed; cut lengthwise into 2 long strips.
5. Bring a large pot of lightly salted water to a boil; slowly lower pasta strips and cook 1 minute 30 seconds, then transfer to a bowl of cold water until cooled; place on work surface and cut each strip in half to make 4 pieces.

Cheese Filling

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

Lasagna

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