Preheat grill to medium. Halve and pit 4 peaches; brush both sides with olive oil. Season with salt and pepper. Place peaches, cut sides down, on grill rack. Cover grill and cook until charred and softened, 4 to 5 minutes. Turn peaches over and continue grilling until skin begins to blister and slide off, 4 to 5 minutes.
For each serving, divide arugula or salad greens among 4 dinner plates. Top each with 2 peach halves, 2 ounces mozzarella slices and 2 ounces ham slices. Drizzle each salad with about 3 tablespoons of Peachy Balsamic Dressing.
To make Peachy Balsamic Dressing, peel and pit 1 peach and place in a blender or food processor with extra-virgin olive oil, balsamic vinegar, honey, Dijon mustard, sea salt and cracked black pepper. Cover and blend or process until smooth.