Heat several inches of vegetable oil in a Dutch oven or other heavy pot with high sides to 350 degrees F. Line a baking sheet with paper towels.

With a sharp knife, cut the ends from each plantain and score the skins lengthwise several times to help remove the skin. Peel the plantains and then cut them crosswise into ¾-inch thick slices. Working in batches, deep fry the plantain pieces until just golden, 2 to 3 minutes. Transfer them to the baking sheet to drain. 

Cut two pieces of parchment paper into 12-inch lengths. Place one piece on a flat work surface and place a few of the oil-blanched plantain slices on the parchment. Cover with the second sheet of parchment, and using the bottom of a heavy pot (such as a cast-iron skillet), press down firmly on the plantains to flatten them to about ¼-inch thickness. Repeat with the remaining slices.

Working in batches, fry the tostones a second time, until they are golden brown and crisp, 2 to 3 minutes. Transfer to the prepared baking sheet to drain briefly. Sprinkle with salt and serve hot or warm.