1 To make the sauce: Heat the oil in a medium saucepan over medium heat. Add the onion and anchovies and cook, stirring occasionally, until the onion is tender, about 4 minutes. Stir in the garlic and hot red pepper and cook until the garlic is fragrant, about 1 minute. Stir in the tomatoes and 1∕2 cup water. Bring to a simmer and reduce the heat to medium low. Simmer, stirring occasionally, until the sauce has reduced slightly but is still fairly liquid, about 20 minutes. (the sauce will reduce more when cooked with the steaks, so no need to cook it more at this point.) Stir in the capers and basil. Season to taste with the salt and pepper. Remove from heat and set the sauce aside.
2 To prepare the steaks: Cut the steak into 8 equal pieces. Using a pointed meat mallet, pound each steak well on both sides to tenderize them. If needed, cut the steaks in half vertically to fit the skillet. Season all over with the salt and pepper.
3 Heat the oil in a very large (at least 12 inches in diameter) skillet over medium-high heat until it is hot but not smoking. In batches without crowding, add the steak and cook, turning halfway during cooking, until seared on both sides, 2 to 3 minutes. Transfer to a platter.
4 Add the sauce and bring to a simmer, scraping up the browned bits in the skillet with a wooden spoon. Return all of the steaks to the skillet and submerge them in the sauce. Bring to a simmer and reduce the heat to low. Cover the skillet and simmer, occasionally turning the steaks, until they are tender, about 50 minutes. If the sauce thickens too much, stir in 1∕4 cup water.
5 Move the steaks in the sauce so they form two layers. Sprinkle with half of the cheese to cover the top layer. Cover with the lid and cook until the mozzarella melts, about 1 minute. Transfer the cheese-topped steaks to a platter. Repeat with the remaining steaks in the skillet and the rest of the mozzarella. Spoon the sauce in the pan around the steaks, sprinkle with the additional basil, and serve.
TIP To shred fresh mozzarella, which is much softer than the industrial low-moisture variety, freeze the cheese for an hour or so to firm it.