In a small bowl, whisk together the basil, garlic, lemon juice and rice vinegar. Gradually whisk in ⅓ cup olive oil. Season with salt and pepper to taste.
Bring a medium pot of salted water to a boil over high heat. Add the green beans and blanch for 1 to 2 minutes. Transfer to a bowl of ice water to cool completely.
Preheat a grill or grill pan over high heat. Brush the remaining olive oil evenly over the peaches and cheese. Working in batches, grill the peaches and cheese until well charred, about 3 minutes per side.
Arrange the mixed greens on a large platter. Top with the green beans, peaches and cheese. Drizzle with the dressing and garnish with the pine nuts.