Melt shortening in a 12-inch cast-iron skillet over medium heat until a frying thermometer registers 375°F.

Meanwhile, whisk together eggs and milk in a bowl. Place flour, cornmeal, salt, and pepper in another bowl; whisk to combine. Place 8 pieces of okra in egg mixture to coat, then dredge in flour mixture. Fry until golden brown, turning as necessary, 3 to 4 minutes.

Drain on paper towels. Repeat coating and frying okra; serve warm.