Pasta Dough

1. Make a pile of flour on a work surface and make a well in the center; add eggs to the well and scramble with a fork, gradually incorporating flour until a shaggy dough forms.
2. Knead dough by hand until a smooth ball, about 3 minutes; cover and rest 10–15 minutes, then knead about 4 minutes more until very smooth and elastic.
3. Divide dough in half; wrap tightly. Shape one piece into a rectangle and roll into a very thin large rectangle, turning and dusting with flour as needed; cut lengthwise into 2 long strips.
4. Bring a large pot of lightly salted water to a boil; slowly lower pasta strips and cook 1 minute 30 seconds, then transfer to cold water to cool; transfer to a 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 the oven to 450 degrees F (230 degrees C).
2. Set aside about 1/2 cup of the cheese mixture; dollop the remaining mixture by spoonfuls over the 4 pasta pieces and spread roughly; spoon about 1/3 cup of sauce onto each piece.
3. Top each piece with torn basil; set aside 3/4 cup mozzarella and scatter the rest evenly over the pasta 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; spread over 1/3 of the reserved cheese mixture, add a few spoons of sauce and a small scatter of shredded mozzarella; repeat with remaining handkerchiefs, then top with sauce, remaining mozzarella, a dusting of Parmesan, and a drizzle of olive oil.
5. Bake until the top is lightly golden and the pan is bubbling around the edges, 25 to 30 minutes; let rest 10–15 minutes, then cut into quarters to serve.