1) Using a vegetable peeler, cut the zucchini into lengthwise ribbons. Peel several from one side until you reach the seeds, then turn the zucchini and peel more until all that is left is the core. (Save the leftover core to use in soup.)
2) Cook the zucchini strips in two batches. Heat 1 tablespoon of the oil in a large nonstick skillet over medium-high heat. When it is hot, add the zucchini ribbons and salt. Cook, tossing and gently stirring the zucchini, for 2 - 3 minutes, until it is softened and begins to turn translucent.
3) Set the batch aside, covering it to keep it warm, and cook the remaining zucchini with the rest of the olive oil.
4) Serve immediately, topped with freshly grated Romano or Parmesan cheese.