Heat oil in a large wide heavy pot over medium-high.

Add pork to pot, breaking apart into 6–8 large chunks with a wooden spoon. Cook, undisturbed, until well browned underneath, about 5 minutes.

Turn pieces and continue to cook, turning occasionally, until pork is browned on 2–3 sides, about 5 minutes longer.

Add ginger, garlic, and sugar to pot and cook, breaking up pork into small clumps, until meat is nearly cooked through, about 5 minutes longer.

Add tomato paste and 2 basil sprigs. Cook, stirring occasionally, until paste darkens, about 2 minutes.

Add chili paste, soy sauce, vinegar, and water. Bring to a simmer, reduce heat to low, and cook, uncovered and stirring occasionally, until sauce is slightly thickened and flavors have melded, 30–45 minutes.

Cook noodles in a large pot of boiling salted water, stirring occasionally, until 1 minute short of al dente. Add to pot with sauce along with butter and a splash of pasta cooking liquid. Simmer, tossing occasionally, until sauce begins to cling to noodles, about 1 minute. Pluck out basil sprigs.

Divide noodles among plates. Top with torn basil.