Place chicken breasts and stock in a large saucepan; bring to a boil over high heat. Skim and discard any foam that rises to the top. Add 1 teaspoon kosher salt and next 3 ingredients. Reduce heat, and simmer 30 minutes or until the chicken is tender and cooked through.
Transfer chicken to a plate. Strain stock through a very fine strainer, discarding solids, and skim off any fat from the top. Set aside.
When chicken is cool enough to handle, remove skin, and pull the meat from the bones. Discard skin and bones, and shred chicken into strips. Set aside.
Heat olive oil in a large stockpot over medium-low heat. Add onion, ground black pepper, and remaining 1 teaspoon kosher salt, and cook, stirring occasionally, 10 minutes. Add bell pepper; cook 5 minutes.
Add tomato, Key lime zest and juice, and reserved shredded chicken and chicken stock. Bring to a boil over high heat. Reduce heat, and simmer about 10 minutes. Divide soup among 6 bowls, and top each with serrano chiles, if desired, and remaining ingredients.