Make the ragu: In a large pot set over medium-high heat, add the oil. When the oil is hot, add the onion, carrot, and celery. Cook, stirring occasionally, until the onion is translucent and beginning to turn golden and soft, 18–20 minutes. Add the pork, season with salt, and cook, stirring frequently, until the meat is no longer pink, about 5 minutes. Add the milk and cook until mostly evaporated, 8–10 minutes. Add white wine and cook until evaporated, 10–12 minutes more. Lightly crush the tomatoes with your hands, then add the tomatoes with their juices and the tomato paste to the pot. Increase the heat to high and bring to a boil, then regulate the heat to maintain a strong simmer. Cook, stirring occasionally, until the meat is tender and the fat in the pot begins to separate and pool on top of the tomatoes, 55–60 minutes.
Meanwhile, make the gnocchi: In a medium pot, add the potatoes and enough water to cover by 2 inches. Bring to a boil over high heat, then regulate the heat to maintain a strong simmer. Cook until the potatoes are tender when poked with a paring knife, about 45 minutes. Drain, discarding the cooking liquid, then rub off the skins with a dry kitchen towel while the potatoes are still hot.
Pass the potatoes through a potato ricer onto a clean, lightly floured work surface. Gradually work in ¾ cup flour with your hands until well incorporated, then knead gently until a soft, smooth dough forms. (Add up to ½ cup more flour as needed to prevent the dough from sticking to the surface.) Divide the dough into 4 even pieces. Cover 3 of the pieces with a dry kitchen towel to keep warm while you shape the gnocchi. 4. Scrape your work surface clean and lightly dust with fresh flour. Roll 1 piece of dough into a ¾-inch-thick log. Cut log crosswise into ¾-inch dumplings, then roll each dumpling against a gnocchi board or the tines of a fork to create ridges. Repeat with remaining pieces of dough. Separate gnocchi into 4 batches.
Fill a large pot two-thirds of the way with water and bring to a boil over high heat. Salt the water generously. Working with 1 batch of gnocchi at a time, cook until they float to the surface, 30–40 seconds. Use a fine-mesh strainer or a large slotted spoon to transfer gnocchi to a warm bowl or serving platter, reserving the cooking water. Repeat with remaining batches.
Adjust the ragu seasoning with salt and pepper as needed. Add the gnocchi and gently fold with a wooden spoon to coat in sauce. (Add ¼–½ cup of the reserved cooking water if the sauce seems pasty.) Transfer to a serving platter, top with Parmigiano-Reggiano and more pepper, and serve immediately.