Bring a small pot of water to a boil. Add chicken; boil until an instant-read thermometer inserted into the center reads at least 165 degrees F (74 degrees C), about 7 minutes. Drain.
Run cool water over chicken to speed cooling process. Shred or finely slice chicken.
Heat olive oil in a large pot over medium heat. Add 1/2 onion, lime juice, 1/2 cup cilantro, jalapeno peppers, and garlic until onion is slightly greenish in color, about 5 minutes.
Combine 4 cups water and bouillon powder in a small bowl. Pour into the pot. Let cook until just heated through, about 5 minutes. Stir chicken into the pot.
Ladle soup into 4 bowls. Top with remaining 1/2 onion, 1/2 cup cilantro, avocado, and queso fresco.
