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Place chicken in medium saucepan of lightly salted water.
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Bring to a boil over med-high heat.
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Reduce to low, simmer until chicken is white throughout but moist. Shred. In medium skillet, heat 1 TBSP oil over med-high heat.
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Add onion and oregano.
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Cook until onion is limp. Remove from heat.
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Stir in cream cheese.
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Add chicken, salt and pepper.
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Mix thoroughly.
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In medium skillet, heat 2 TBSP oil over med-high heat.
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Using tongs, soften tortillas 1 at a time by dipping in hot oil for about 2 seconds on each side.
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Add oil if needed.
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Drain between paper towels.
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Stack and keep covered with a clean towel.
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To fill: Place 1 tortilla on work place, spoon about 2 TBSP mixture in strip in lower 1/3 of tortilla.
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Roll up tightly.
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Secure with toothpick and lay seam side down on plate.
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Repeat with all tortillas.
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Taquitos can be held, covered and refrigerated, up to 6 hours before frying. Cover with plastic wrap to keep from drying out.
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In medium skillet, heat remaining oil over med-high heat.
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Fry taquitos, 2 at a time, seam side down, turning, until lightly browned on both sides, about 2 minutes.
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Drain on paper towels.
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Cut each taquito in half.