Heat half the olive oil in a large nonreactive skillet. Add the butter and as soon as it melts, add the scallions and red onion. Cover and sweat over low heat just until soft, about 4 minutes.
Add the potatoes, stir, and cook uncovered over medium heat for another 3 minutes, stirring the vegetables several times with a spatula.
Add the pasta,
Cut the basil leaves into strips about
Sprinkle the cheese over the pasta and mix it in gently. Add half the remaining olive oil and the cut basil leaves and mix in delicately. Check for seasoning. Divide among 4 heated open soup plates. Drizzle the remaining olive oil on top, add a grinding of pepper, and serve immediately.