Heat 1/4 cup oil in a 10-inch sauté pan. Add butter. When it melts, add onions and potatoes. Cook, stirring gently, over medium heat until they begin to turn golden.

In a small saucepan, bring stock to a slow simmer. Add pasta to sauté pan, and stir gently. Lightly season with salt and pepper, and add tomatoes, garlic and basil or arugula stems. Add 1 1/2 cups stock. Cook, stirring gently, until nearly all stock has evaporated. Add scallions and another cup of stock, and cook, stirring, adding additional stock from time to time, so there is always some liquid in the pan, until pasta is al dente, about 18 minutes. Remove garlic and herb stems.

Fold in cheese and all but 1 tablespoon remaining oil. Add slivered herbs. Season with additional salt and pepper if needed. Transfer to warm soup plates, taking care that the ingredients are well distributed. Drizzle remaining oil over each and serve.