Heat gas or charcoal grill. Sprinkle chicken with salt and pepper.
Carefully brush oil on grill rack. Place chicken on grill over medium-high heat. Cover grill; cook 12 to 15 minutes, turning once, until juice of chicken is clear when center of thickest part is cut (at least 165°F). Let stand 5 minutes.
Meanwhile, in medium bowl, toss strawberries, nuts and spinach. In small bowl, stir mayonnaise, vinegar, honey and poppy seed with whisk until well blended.
Thinly slice chicken. On 4 plates, evenly divide spinach mixture, chicken and cheese. Drizzle with dressing.