How to Make It
In a medium saucepan, combine the rice vinegar, tamari, fish sauce, ginger, chiles, sugar and 1/2 cup of water and bring to a simmer over moderate heat. Let cool slightly, then add the tomatoes. Let stand at room temperature for 2 hours. Drain the tomatoes; discard the ginger and chiles.
In a standing mixer fitted with the paddle, combine the eggs, egg yolks and 2 tablespoons of the oil. Add the 2 cups of flour and 1 1/2 teaspoons of salt and mix until the dough comes together. Knead the dough on an unfloured surface for 30 seconds, then wrap in plastic and let stand at room temperature for 1 hour.
Cut the dough into 2 pieces. Working with 1 piece at a time and keeping the other covered, roll the dough through successively narrower settings on a pasta machine until the pasta is very thin. Dust with flour and let stand on a work surface, turning the pasta once or twice, until slightly dry, about 30 minutes. Tear the pasta into 2-inch pieces and toss with flour.
Bring a large pot of generously salted water to a boil. Add the pasta and cook until al dente, about 3 minutes. Drain, reserving 1/2 cup of the cooking water.
In a large skillet, heat the remaining 1 tablespoon of grapeseed oil. Add the ginger and onions and cook over moderate heat until softened, about 2 minutes. Add the 1/2 cup of pasta cooking water and bring to a boil. Add the butter and simmer over low heat, shaking the pan, until the sauce is creamy. Add the pasta, red chile slices, basil and mint and toss until evenly coated. Add the goat cheese and pickled tomatoes and serve right away.