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 of the onion, 1/2 cup cilantro, jalapeño peppers, lime juice, and garlic; cook 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. Cook until just heated through, about 5 minutes. Stir in chicken.

Ladle soup into four bowls. Top with remaining avocado, onion, cilantro, and queso fresco.