In a large mixing bowl, whisk together 2 eggs, 1 egg yolk, Pecorino Romano, 1/2 cup Parmesan, 1 tablespoon water, 1/2 teaspoon kosher salt, and freshly ground black pepper until smooth and cohesive. Set this egg-cheese mixture aside.
Fill a large pot with water and bring it to a boil over medium-high heat. Generously salt the water. Meanwhile, prepare a medium bowl of ice water. Once the pot of water is boiling, carefully lower the additional 2 eggs into the pot with a slotted spoon or tongs. Boil these eggs until hard-boiled, around 10 minutes, then transfer to the ice water to cool. After they have cooled, peel and reserve for later.
Add the rigatoni to the same pot of boiling water, cooking until al dente per package instructions. Reserve 1/2 cup of the pasta cooking water, drain the pasta, and set it aside.
In a large nonstick skillet, melt 1 tablespoon of butter over medium heat. Add the panko breadcrumbs, stirring frequently, and toast until golden brown, approximately 5 minutes. Add parsley, lemon zest, and the remaining 2 tablespoons of Parmesan. Stir to combine, then transfer the mixture to a small bowl and set aside.
Wipe the skillet clean and return it to the stove over medium-high heat. Melt the remaining 2 tablespoons of butter, then add the sliced garlic. Cook, stirring occasionally, until garlic is softened and fragrant, about 1-2 minutes.
Add the drained rigatoni to the skillet, tossing to coat in the butter and garlic mixture. Reduce the heat to medium-low. Pour in 1/4 cup of the reserved pasta water and add the egg-cheese mixture, stirring constantly with a rubber spatula to prevent scrambling. Continue cooking and stirring for 2-4 minutes, allowing the sauce to become smooth and creamy. Add more pasta water as needed to reach the desired consistency. Adjust salt and pepper to taste.
Serve by spooning the pasta into four bowls, topping each with the toasted panko mixture. Using a fine grater, grate the hard-boiled eggs over the pasta to create a delicate, airy topping. Enjoy immediately.
