1. In a small bowl, stir together papaya, red onion, jalapeno pepper, cilantro, lime juice, and 1/4 teaspoon of salt. Refrigerate salsa at least 15 minutes or up to 4 hours.
2. Rub chicken all over with oil, then season with thyme, cayenne, and remaining 1/4 teaspoon salt. Let stand at room temperature 15 minutes.
3. Prepare a hot fire in a barbecue grill. Grill chicken, turning once or twice, until white throughout, 5 to 7 minutes. Serve grilled chicken with salsa spooned on top