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In a wide, shallow dish or pie plate, whisk together flour, cornstarch, 2 teaspoons salt, and 1/4 teaspoon pepper. Transfer 1 1/4 cups flour mixture to an airtight container; set aside. In a large glass bowl, whisk together buttermilk, 2 tablespoons salt, and 3/4 teaspoon pepper. Dredge chicken in flour mixture, then submerge in buttermilk mixture. Cover and refrigerate 3 hours (or up to overnight).
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Place a wire rack on a rimmed baking sheet lined with paper towels. In a large cast-iron or other heavy skillet, heat oil to 350 degrees over medium (a small cube of bread should brown in less than 1 minute). Transfer reserved flour mixture to a wide, shallow dish. In batches, lift chicken from buttermilk, letting excess drip off, and dredge in flour mixture.
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Fry chicken until golden brown and cooked through, 16 to 20 minutes per batch, flipping once (adjust heat if browning too quickly). With tongs, transfer chicken to rack to drain, 5 minutes, then serve.