The broth flavoring. In a medium-size skillet, heat 1 tablespoon of vegetable oil over medium heat. Add the onion and whole garlic cloves, and fry until both are a deep golden-brown, 12 to 15 minutes. Scoop into a blender, add the tomatoes and blend until smooth.
The broth. Stir the broth into the tomato mixture, partially cover and simmer for 1/2 hour over medium heat. Season with salt.
The garnishing ingredients. If the tortillas are fresh or moist, let them dry out for a few minutes in a single layer. Slice them in half, then slice the halves crosswise into 1/4 inch strips. Heat 1/3 cup vegetable oil in a medium-small skillet over medium-high. When hot, add the tortilla strips and fry, turning frequently, until they are crisp. Drain on paper towels. Cut the chiles into 1-inch squares and fry in the hot oil very briefly, about 3 or 4 seconds; immediately remove and drain, then place in a small serving bowl.
Assembling the soup. Just before serving, divide the cheese among 4 to 6 bowls, and top with the fried tortilla strips. Ladle on the hot soup and serve right away. Pass the toasted chile separately, along with the lime.
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