Combine bulgur and water in a large bowl. Cover and let stand 30 minutes.
Heat oil in a nonstick skillet over medium-high heat. Add asparagus, peas, and onion; sauté 4 minutes. Add asparagus mixture, chicken, cilantro, and parsley to bulgur mixture.
Combine broth and next 4 ingredients (broth through pepper). Pour over bulgur mixture; toss to coat. Arrange greens on each of 4 plates; top with bulgur mixture.