Fill a large cast-iron skillet with 1 inch of vegetable oil. Heat over medium heat until a deep-fry thermometer registers 350 degrees F.
Meanwhile, whisk the flour, cornmeal, cornstarch, thyme, 1 tablespoon salt and 2 teaspoons pepper in a large bowl. Whisk the eggs and milk in another large bowl. Season the chicken all over with salt and pepper. Dredge each piece in the flour mixture, then dip in the egg mixture, letting any excess drip off. Return the chicken to the flour mixture and turn until well coated. Transfer to a large plate or baking sheet.
Carefully add about half the chicken to the hot oil (do not crowd the skillet). The oil temperature will drop; adjust the heat as needed to maintain a temperature of 350 degrees F. Fry the chicken, without turning, until the crust starts browning, about 2 minutes. Flip the chicken and continue cooking, turning occasionally, until browned and crisp all over and a thermometer inserted into the breasts registers 160 degrees F and the thighs 170 degrees F, 13 to 16 minutes. Remove with tongs and transfer to a rack set on a baking sheet; sprinkle with salt. Let the oil temperature return to 350 degrees F before frying the remaining chicken.