Cut each chicken breast crosswise into 1-1/2-inch-wide pieces.Heat the oil in a 6-quart pot over medium-high heat until shimmering. Add the onion and cook, stirring often, until softened, about 5 minutes. Stir in the garlic and jalapeño and cook, stirring often, until fragrant, about 45 seconds. Add the broth, hominy, oregano, and chicken. Raise the heat to high and bring to a boil. Reduce the heat to medium, cover, and simmer gently, stirring occasionally and adjusting the heat as needed to maintain a simmer, until the chicken is cooked through, about 10 minutes. Transfer the chicken to a plate. Using two forks, shred the meat into bite-size pieces and return to the pan. Bring the soup back to a simmer over medium heat, stir in the lime juice, and season to taste with salt and pepper. Ladle into bowls, top with the cheese, and serve immediately.