Line a plate with paper towels. Heat ¼ to ½ inch of vegetable oil in a skillet or Dutch oven (or you can use a deep fryer). When the oil reaches 350 degrees, place a few of the tortilla strips in at a time to cook. (Be careful not to overcrowd the pot.)
Fry the strips for just a couple of minutes, until they start to turn light brown, remove from the oil and let drain on paper towels. Set aside.
To poach the chicken:
In a large pot or Dutch oven over medium-high heat, bring chicken broth, 2 onion quarters, 2 garlic cloves, cilantro, oregano, 1/2 teaspoon salt, and 1/4 teaspoon pepper to a boil.
Add chicken breasts, reduce heat to low, cover, and simmer until chicken is cooked and registers 165 degrees on a digital thermometer, about 15 to 20 minutes.
Remove chicken to a cutting board and shred it into bite-sized pieces, discarding bones and skin. Strain broth through a fine-mesh strainer, discarding solids. Reserve broth and set aside.
To make the soup base:
While the broth is simmering, adjust an oven rack to 4 to 6 inches from the heat source and preheat broiler on HIGH. Line a baking sheet with foil for easy clean-up.
Arrange tomatoes, remaining 2 onion quarters, 2 garlic cloves, and tomatoes on the prepared sheet. Broil until vegetables are charred and sizzling, about 15 minutes, rotating pan every 5 minutes to ensure even broiling.
Transfer broiled vegetables to a blender. Add chipotle pepper. Process until smooth, about 30 seconds.
Add 1 tablespoon oil to the empty pot and heat until shimmering over medium-high heat. Add blended tomato mixture and 1/4 teaspoon salt. Cook, stirring frequently, until mixture is darkened and liquid is evaporated, about 10 minutes.
To make the tortilla soup:
Stir broth into tomato soup base and simmer until flavors meld, about 15 minutes, scraping any browned bits from the bottom of the pot.
Stir in shredded chicken. Continue to simmer until chicken is heated through, about 5 minutes. Remove from heat and season to taste with salt and pepper.
Place a small amount of tortilla strips in the bottom of each bowl. Ladle soup over the top. Serve with cheese, sour cream, avocado, cilantro, lime, and tortilla strips, or your favorite toppings.
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Notes
Corn tortillas: Store-bought tortilla chips are too flimsy and bland for this soup. You can use store-bought corn tortillas, then slice and fry them yourself.
Cilantro: If you hate cilantro, just leave it out.
Mexican oregano: This dried herb tastes more like marjoram than Italian oregano, and you can find it in the Hispanic food aisle at some grocery stores. Substitute Italian oregano if that’s what you have.
Chipotle chiles: These canned chiles in adobo sauce are spicy! If you shy away from spicy food, add a small amount to start or leave them out.
Crema: Mexican crema is a lot like sour cream in flavor but with a thinner consistency. It's ideal for drizzling whereas sour cream is easier to dollop.
Yield: This recipe makes 8 cups of soup, enough for 4 hearty servings, 2 cups each.
Storage: Store leftovers covered in the refrigerator for up to 4 days.