01 - Preheat grill or grill pan to medium-high heat.
02 - Lightly brush peach halves with 1 tablespoon olive oil. Grill peaches cut side down for 2-3 minutes, until grill marks form and fruit softens slightly. Remove and let cool, then slice.
03 - In a small bowl, whisk together remaining olive oil, lime juice, honey, Dijon mustard, salt, and pepper to make the dressing.
04 - In a large salad bowl, combine arugula, grilled peach slices, avocado, red onion, and basil. Toss gently.
05 - Drizzle salad with dressing and toss to coat. Top with toasted pecans or walnuts and crumbled cheese, if using.
06 - Serve immediately.