Simmer the chicken breast gently in the broth until cooked through and no longer pink in the center, about 15 to 18 minutes. Lift it out, shred it with two forks, and keep the broth.
While the chicken cooks, char the whole tomatoes, the halved onion and the garlic in a dry pan over high heat until blackened in patches. Toast the stemmed, seeded ancho chile in the same pan for a few seconds a side, just until it smells fragrant, then soak it in a little of the hot broth to soften.
Blend the charred vegetables, the softened ancho and a cup of the broth to a smooth puree.
Heat the oil in the soup pot over medium-high until shimmering, then pour in the puree and fry it, stirring, for 4 to 5 minutes until it darkens and thickens. This step, frying the blended salsa, is what separates a deep tortilla soup from a thin one.
Add the remaining broth, the cumin and salt, bring to a simmer, and cook 15 minutes. Stir the shredded chicken back in and simmer 5 minutes more.
For the strips, slice the corn tortillas thin and fry them in a shallow layer of oil over medium-high heat in small batches, 30 to 45 seconds until crisp and just golden, then drain on paper. Serve the soup topped with the strips.