Snap or cut off tough ends of asparagus spears; place spears in shallow glass dish. Drizzle with 1 teaspoon of the oil; turn to coat. Cover; refrigerate until grilling time.
In small bowl, mix remaining oil, the garlic, rosemary and seasoned salt. Rub both sides of each chicken breast with oil mixture; place in another shallow dish. Cover; refrigerate 30 minutes to marinate.
Heat gas or charcoal grill. Place chicken on grill over medium heat. Cover grill; cook chicken 15 to 20 minutes, turning once or twice and adding asparagus after 7 minutes, until juice of chicken is clear when center of thickest part is cut (170°F). Cook asparagus 6 to 8 minutes, turning frequently, until crisp-tender.
Cut asparagus into 1-inch pieces. In large bowl, toss asparagus, salad greens, tomatoes and onion. Pour dressing over salad; toss to coat. Cut chicken into strips; serve over salad.