Cut the tortillas into 1/4-inch strips. Heat the oil in a medium skillet and, when very hot, fry the tortilla strips, in small batches, until lightly golden and crisp, 30 seconds to 1 minute. Transfer to paper towel lined plate to drain. Season with salt, to taste. Repeat until all tortilla strips have been fried. Set fried tortilla strips aside and reserve the vegetable oil.
Transfer 1 tablespoon of the reserved cooking oil to a large saucepan and add the chopped onion, celery, carrot, and jalapeno pepper. Cook over medium-high heat, stirring occasionally, until vegetables have softened, about 4 minutes. Add the garlic, bay leaf, and Mexican oregano and cook, stirring, for 1 minute. Add the tomato and season lightly with salt. Cook, stirring, until the tomato is softened and has released its liquid and the mixture is nearly dry, 4 to 5 minutes. Add the chicken stock and chicken breasts and bring to a boil. Reduce heat to a slow simmer and cook until the chicken is just cooked through, 12 to 15 minutes. Remove chicken from the soup and set aside until cool enough to handle. Allow soup to continue simmering.
When the chicken has cooled a bit, shred into bite size pieces and return to the pot along with the green onions and lime juice. Cook for 5 minutes, or until the chicken is heated through and the soup is piping hot. Season the soup, to taste, with salt and ladle the soup into wide soup bowls, with a handful of tortilla strips added to each bowl. Garnish with the avocado and cilantro and serve immediately.