Bring 4 quarts of water to a boil in a large pot and add 2 tablespoons salt. Add the eggnoodles and cook according to the package directions, about 7 minutes for al dente. Drain, reserving ½ cup of the pasta cooking water.
Combine the butter with the pepper in a large skillet. Place over medium heat and cook, stirring, until the butter is melted and bubbling, about 1 minute. Add the reserved pasta cooking water and bring to a simmer. Reduce the heat to low and add three-quarters of the Pecorino Romano. Whisk until smooth, then add the pasta to the pan. Increase the heat to medium and stir gently with a heat-resistant spatula until a sauce forms and sticks to the pasta. Add the parsley and toss to coat the pasta; season with salt and more black pepper to taste.
Divide the pasta among 4 bowls and sprinkle with the remaining Pecorino Romano.
Tip: If the sauce begins to separate and look greasy or lumpy, add more of the pasta cooking water (cottura) and bring back to a simmer until smooth.