Put the potatoes in a medium pot and cover with cold water by 1 inch. Bring to a simmer over medium heat and cook until easily pierced with a knife, 20 to 25 minutes. Drain and let cool slightly. Peel the potatoes and pass through a ricer onto a clean surface in a thin layer (do not mound the riced potatoes). Let cool completely, at least 10 minutes.
Combine 1 teaspoon salt, the finely ground pepper and the nutmeg in a small bowl; sprinkle over the riced potatoes. Drizzle the beaten egg over the potatoes, then sprinkle with 1 cup flour and 1/2 cup pecorino. Chop the potato mixture with a bench scraper, turning the mixture onto itself until it just starts to come together. Lightly dust the dough with flour and knead, adding up to 1/4 cup more flour if needed, until the dough is almost smooth and slightly wet but not sticky. Form the dough into a ball.
Cut the dough into 4 pieces using the bench scraper; keep covered with a kitchen towel until ready to use. Cut 1 piece of dough in half; dust with flour and gently roll each half into a 1/2-inch-thick rope. Cut each rope into 1/2-inch pieces with the bench scraper or a sharp knife and dust with flour. Repeat with the remaining dough.
Line 2 baking sheets with clean kitchen towels; dust with flour. Lightly flour your hands and roll each piece of dough between your palms to make a small ball. One at a time, place each dough ball on the tines of a fork closest to the handle and hold it in place with a floured thumb. In one gentle motion, quickly press and roll the dough along the tines toward the tip of the fork. Transfer to the prepared baking sheets. (The uncooked gnocchi can be frozen at this point. Freeze on the baking sheets until firm, then transfer to a resealable plastic bag and freeze up to 3 weeks.)
Preheat the broiler and bring a large pot of salted water to a boil. Melt the butter in a large ovenproof skillet over medium heat. Stir in the coarsely ground pepper and cook until toasted, about 30 seconds. Add 3/4 cup of the boiling water. Bring to a boil and cook until reduced by half, 2 to 3 minutes. Remove from the heat.
Working in batches, shake the gnocchi in a colander to remove any excess flour, then add to the pot of boiling water and cook until the gnocchi float to the top, about 2 minutes. Remove the gnocchi with a slotted spoon or spider and transfer to the skillet with the butter sauce, off the heat. Once all the gnocchi are in the skillet, stir to coat and return the skillet to medium heat. Bring the sauce to a boil, adding up to 1/4 cup of the cooking water if the gnocchi seem dry. Remove from the heat and stir in 1/4 cup each pecorino and parmesan. Sprinkle the remaining 1/4 cup each pecorino and parmesan over the top. Transfer the skillet to the broiler and broil until browned and bubbling, 1 to 2 minutes.