To make the dressing, in a small bowl, combine the mayonnaise, buttermilk, dill, garlic and black pepper. Whisk until smooth. Cover and refrigerate until needed.
Prepare a hot fire in a charcoal grill or heat a gas grill or broiler (grill). Away from the heat source, lightly coat the grill rack or broiler pan with cooking spray. Position the cooking rack 4 to 6 inches from the heat source.
Brush the chicken breasts with olive oil. Grill or broil until browned and just cooked through, about 5 minutes on each side. Transfer the chicken to a cutting board and let rest 5 minutes before slicing into strips.
In a large bowl, combine the salad greens, bell pepper, onion and most of the dressing. Toss to mix well.
Transfer the salad to individual plates. Add the sliced chicken and drizzle with the remaining dressing. Serve immediately.