Preheat a gas or stove-top grill.

Cut squash lengthwise into ¼-inch slices. Brush both sides with 1 tablespoon oil and season to taste with salt and pepper. Grill for about 2 minutes per side, until nicely marked and tender but still firm. Remove to a cutting board and cut crosswise to form 1- to 1½-inch squares. Place in a large bowl. Cut roasted bell peppers into squares roughly matching squash; add to bowl.

Pound chicken breasts to an even thickness. Combine 1 tablespoon oil and 1 tablespoon vinegar; brush on both sides of chicken. Season to taste with salt and pepper. Grill until edges turn opaque, about 3 minutes. Turn and cook another 2 minutes, or until cooked through but still tender. Remove to a plate, cover, and keep warm

Whisk remaining ¼ cup oil, 2 tablespoons vinegar, mustard, salt, and pepper in a small bowl until emulsified. Pour over squash and peppers. Add shallots and toss well. Add arugula and toss gently but thoroughly. Divide salad among four plates. Cut chicken across the grain into thin slices and arrange on top of salads. Garnish with parsley and serve.