Chilaquiles are a Mexican dish usually eaten for breakfast. Look for dried ancho chilies in Latino markets. For a milder sauce, omit the jalapeño chili.
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1. In a 3- to 4-quart pan over high heat, combine 1 1/2 quarts water, onion, and garlic. Rinse dried ancho chilies, break off and discard stems, and add chilies to pan. Cover and bring to a boil. Add chicken, cover, and return to a boil. Remove from heat and let stand, covered, until chilies are soft and chicken is no longer pink in center of thickest part (cut to test), 12 to 18 minutes. If chicken is still pink, return it to the hot liquid, cover pan, and let steep a few minutes longer.
2. Meanwhile, cut tortillas into 1/2-inch-wide strips, then cut strips in half crosswise. Put oil in a 10- to 12-inch frying pan over medium-high heat. When oil is hot, add a fourth of the strips and cook until lightly browned and crisp, 3 to 5 minutes. Remove with a slotted spoon and drain on paper towels. Keep warm in a 200° oven. Repeat to fry remaining strips.
3. Remove cooked chicken from liquid, reserving liquid and vegetables. When chicken is cool enough to handle, in about 10 minutes, tear meat into shreds.
4. Pour onion-chili mixture into a strainer set over a bowl; reserve liquid and put strained vegetables into a blender or food processor. Purée chili mixture, tomatoes, and jalapeño until smooth. If you prefer a thinner sauce, add 2 to 3 tablespoons reserved liquid (discard remainder). Season to taste with salt.
5. Arrange tortilla strips on a platter or plates; cover with eggs, chicken, chili sauce, cheese, and cilantro. Garnish with sour cream and lime wedges.
If making chicken and sauce up to 1 day ahead, cool, cover, and chill. Reheat chicken, covered, in a microwave oven at half power (50%), stirring once, until warm, about 4 minutes; reheat sauce in a microwave oven on full power (100%), stirring once, until hot, about 2 minutes. If frying tortilla strips up to 1 day ahead, wrap airtight; reheat, uncovered, in a 200° oven until warm, about 5 minutes.
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