How to Make It
Preheat grill to medium-high (about 450°F). Brush chicken with 1 tablespoon oil; sprinkle evenly with 1/4 teaspoon salt and 1/4 teaspoon pepper. Place chicken on grill grate coated with cooking spray. Grill, uncovered, until a meat thermometer inserted in thickest portion registers 160°F, 5 to 6 minutes per side. Let stand 5 minutes; cut into slices.
Whisk together remaining 3 tablespoons oil, vinegar, honey, mustard, remaining 1/2 teaspoon salt, and remaining 1/4 teaspoon pepper in a large bowl. Add arugula, basil, and onion; toss gently to coat. Divide salad evenly among 4 plates; top evenly with chicken, blackberries, and cheese.