Melt shortening in a 12-inch cast-iron skillet over medium heat until a frying thermometer registers 375 degrees. Whisk eggs and milk together in a bowl. Place flour, cornmeal, salt and pepper into another bowl. Whisk to combine. Place 8 pieces of okra in egg mixture to coat, then dredge through flour mixture. Fry until golden brown, turning as necessary, 3 to 4 minutes. Drain on a paper towel. Repeat process. Serve warm.