Remove mushroom stems. Thinly slice caps; set caps aside. Bring stems, broth, and next 6 ingredients to a boil in a saucepan. Remove pan from heat.
Bring 1 quart of water to a boil in a large saucepan. Add noodles; cook 3 minutes or until done. Drain.
Sprinkle chicken with black pepper. Place seeds in a dish. Press seeds into both sides of chicken. Heat a nonstick skillet over medium-high heat. Add oil; swirl to coat. Add chicken; cook 3 minutes on each side or until done. Remove from pan. Add bok choy, cut sides down; cook 3 minutes or until browned. Add reserved mushroom slices and bell pepper. Strain broth mixture through a sieve into pan; cover and cook 2 minutes. Remove vegetables with a slotted spoon.
Thinly slice chicken. Place 1/2 cup noodles, about 1/4 cup vegetables, and 4 ounces chicken into 4 shallow bowls. Spoon 1/4 cup broth mixture over each. Garnish each with two lime wedges.