1. In a large frying pan, heat 1 tablespoon of the oil over moderate heat. Add the tortilla strips and cook, stirring occasionally, until brown and crisp, 3 to 4 minutes. Remove the strips from the pan; drain on paper towels. Toss the hot tortilla strips with 1/4 teaspoon of the salt.
2. In the same pan, heat 2 tablespoons of the oil over moderate heat. Season the chicken breasts with 1/4 teaspoon of the salt, the black pepper, and the cumin and add to the pan. Cook until brown, about 5 minutes. Turn and cook until almost done, about 3 minutes more. Cover the pan, remove from the heat, and let steam for 5 minutes. Remove the chicken from the pan. When the chicken breasts are cool enough to handle, cut them into 1/4-inch-thick slices.
3. Meanwhile, in a large glass or stainless-steel bowl, whisk together the lime juice, cayenne, soy sauce, and 1/4 teaspoon of the salt. Add the remaining 5 tablespoons of oil slowly, whisking. Set aside 2 tablespoons of the vinaigrette.
4. Put the lettuce, the tomatoes, and the remaining 1/4 teaspoon salt into the bowl with the vinaigrette and toss. Put the salad on plates. Top with the chicken. Drizzle the reserved vinaigrette over the chicken and sprinkle the tortilla strips over all. Serve with the lime wedges.
Wine Recommendation: A spicy but crisp red wine will work well with this robust salad. For something a bit offbeat, look for a Sancerre rouge, made from pinot noir, or a pinot noir from Alsace.