Place chicken and 1/2 teaspoon salt in a small saucepan and cover with water; bring to a boil. Immediately remove from heat, cover, and let stand 15 minutes. Drain chicken and run under cold water. Cool completely; set aside. Using 2 forks, shred into bite-size pieces; reserve.
Meanwhile, use a vegetable peeler to remove outer layer of broccoli stems; discard. Continue peeling stems into long strips. In a large bowl, whisk together oil, lemon juice, and 1/4 teaspoon each salt and pepper. Add broccoli strips, Brussels sprouts, celery, hazelnuts, parsley, and reserved chicken to bowl with dressing; toss. Divide among 4 shallow bowls. Top with cheese.