Make the filling. Preheat an oven to 300°F/150°C. In a large bowl, combine chopped garlic, ½ cup shredded Provolone, ½ cup grated Pecorino, ¼ cup breadcrumbs, ¼ cup chopped parsley, 2 Tbsp basil, 2 Tbsp olive oil, ½ tsp Kosher salt, and ¼ tsp black pepper. Stir to mix well.
Assemble the braciole. Place a piece of meat in between a layer of plastic wrap, then use the toothed side of a meat mallet to pound out until about ¼” thick. Repeat with remaining pieces of steak, then place on a large cutting board or work surface for easy assembly. Lay 1 slice of prosciutto on each piece of meat. Divide the filling evenly between slices, spreading into an even layer on top of the prosciutto and leaving a 1-inch border around the edges. Roll each piece of meat tightly so that the filling is completely enclosed, then secure either with kitchen twine or a toothpick to seal. Season the outside of each braciole lightly with Kosher salt and black pepper.
Brown the braciole. Heat 2 Tbsp (30ml) neutral oil over medium-high heat in a wide, deep pan. When hot, add the braciole and cook, browning on all sides, about 8-10 minutes total. Transfer to a plate.
Make the sauce. Add 3 cloves thinly sliced garlic, then cook until fragrant and light golden brown, about 1 minute, then deglaze with 1 cup dry white wine. Bring to a boil, then use a spatula to scrape any browned bits from the bottom of the pan, cooking for about 1-2 minutes. Add 1 cup low-sodium chicken stock, 1 can of crushed tomatoes, and basil leaves, then stir.
Braise until tender. Return the beef braciole to the sauce, along with any drippings on the plate. Cover loosely with aluminum foil, then transfer to a preheated oven and cook for 1½ hours, until the meat is very tender. Baste the meat with sauce every 30 minutes, adding a bit of water or stock as needed if the liquid reduces too much.
Slice, then serve. Serve beef braciole whole with a spoonful of sauce or transfer each roll to a cutting board, thinly slice into rounds, then serve with toasted pine nuts (optional) and fresh basil or parsley.