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In a large bowl, mix together 1 gallon of water and 1 cup salt; stir until salt is dissolved. Add chicken pieces and let stand, refrigerated, 8 to 12 hours.
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Drain chicken and rinse out bowl. Return chicken to bowl and pour over buttermilk. Cover bowl and refrigerate 8 to 12 hours. Drain chicken on a wire rack set over a parchment paper-lined baking sheet, discarding buttermilk.
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Meanwhile, place the lard, butter, and country ham into a large heavy skillet. Cook over low heat, skimming as needed, until butter has stopped foaming and country ham is browned, 30 to 45 minutes. Using a slotted spoon, remove ham and discard. Increase heat to medium-high and heat fat until it reaches 335 degrees on a deep-fry thermometer.
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In a shallow bowl, mix together flour, cornstarch, remaining 1 teaspoon salt, and pepper. Gently squeeze chicken to remove any excess buttermilk. Dredge chicken pieces thoroughly in the flour mixture, shaking off any excess.
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Working in batches, gently place chicken pieces in the heated fat. Cook until chicken is golden brown and cooked through, 8 to 10 minutes. Transfer to a wire rack set over a parchment paper-lined baking sheet or crumpled paper towels. Serve fried chicken hot, warm, room temperature, or cold.