- Using blender or food processor, chop basil and garlic. Slowly drizzle in olive oil and vinegar and process until smooth. Season to taste with kosher salt and freshly ground black pepper.
- Preheat grill to medium-high heat. Toss scallions with oil and season with salt. Grill until charred, turning often, about 2 minutes. Grill peaches and avocados, flesh side down until charred and tender, about 1 minute.
- Arrange greens on large platter. Peel avocado, slice peaches and avocado, and arrange over greens. Top with scallions and drizzle with desired amount of dressing. Shave cheese with a vegetable peeler and sprinkle over salad. Serve immediately.