How to Make It
Preheat grill to medium-high. Whisk together 4 Tbsp. oil, vinegar, shallot, mustard and 1/2 tsp. salt.
Brush peaches with 1 tsp. oil. Grill, flat side down, until warmed through, about 4 minutes. Remove from grill.
Brush chicken on both sides with remaining 2 tsp. oil and sprinkle with salt. Grill until cooked through, turning once, about 8 minutes total. Cut chicken into thin diagonal slices. Cut peaches into 1/2-inch-thick slices.
In a large bowl, toss greens with dressing. Divide greens among serving plates, top with peach halves and sliced chicken, and serve.