Heat oil in a dutch oven over medium-high heat. Add onion, and saute for 2-3 minutes. Lightly sprinkle both sides of chicken breasts with salt and pepper. Place chicken on top of the cooking onions, and cook about five minutes per side.
Pour broth and water over the chicken. Add slices of ginger, and squeeze in lemon juice. Bring broth to a low boil.
Cover and reduce heat to medium. Cook 15 minutes, flipping chicken once. Remove cover, and carefully remove cooked chicken pieces to a cutting board. Shred chicken with two forks, and chop long pieces into smaller bite-sized pieces. Return chicken to the pot.
Fill bowls with washed bean sprouts. Ladle hot soup over the sprouts. Serve with additional lemon wedges and Sriracha hot sauce to taste.
If you prefer, cook rice noodles according to package directions, and serve soup over noodles instead.