Heat oil in a large Dutch oven over medium-high heat. Add 1-inch tortilla pieces; cook 2 minutes or until crisp, stirring occasionally. Add the onion, jalapeño, and garlic; sauté 3 minutes. Add tomato paste and tomatoes; bring to a simmer, and cook 10 minutes. Stir in cumin and broth; bring to a boil. Reduce heat to medium, and simmer, uncovered, 40 minutes or until reduced to 8 cups.
Place half of soup in a blender; process until smooth, and pour into a bowl. Repeat procedure with remaining soup. Return puréed soup to pan. Cook over medium-low heat until thoroughly heated.
Arrange tortilla strips in a single layer on a baking sheet. Bake at 400° for 7 minutes or until crisp; set aside.
Divide chicken and avocado evenly among 8 soup bowls. Pour soup into each bowl, and top with cheddar cheese, cilantro, and tortilla strips.
This soup has a good, but mild flavor. I think next time I make it I will add a little crushed red pepper. Instead of using whole tortillas, I just used tortilla chips which seemed much easier. We served this with a salad and "featherweight buttermilk biscuits" from Bon Appetit 2/98.
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