Cut the plantains into 1-inch pieces on the diagonal. Put about 1/8 inch of oil in a large skillet over medium heat. When it’s hot, add the plantains, without crowding them, and cook, turning as necessary and adjusting the heat so the plantains brown slowly without burning; you may need to do this in batches. Be especially careful as they near doneness; there is so much sugar in the plantains that they burn easily. The process will take 10 to 15 minutes.
Sprinkle with salt, pepper, and lime juice and serve hot.