Place a large pot of water on the stove to boil. Salt the water, and then cook the pasta to al dente, reserving 1 cup of cooking liquid before draining.
Meanwhile, fill a large bowl or sink with cold water, slice the leeks lengthwise, and then slice into 1/2-inch thick pieces. Wash the leeks, separating all the layers, and drain on a kitchen towel.
Heat the extra-virgin olive oil and butter over medium heat, add the leeks, and season with salt and pepper. Cook until the leeks are tender and soft, about 8 to 10 minutes. Add the wine and stir. In a small bowl, beat the egg yolks and temper with the reserved hot starchy cooking water. Add the tempered eggs to the sauce. Add the pasta to the leek sauce and toss to combine for 2 minutes, or until a sauce forms and coats the pasta. Top with Parmigiano-Reggiano cheese and serve.