Heat the oven to 300°F (150°C). In a small skillet over medium heat, toast the breadcrumbs with 1 tablespoon of the filling olive oil, stirring, until just golden, 2–3 minutes. Transfer to a bowl and cool slightly.
Stir the Parmesan, parsley, minced garlic, lemon zest, remaining 2 tablespoons filling olive oil, 1/2 teaspoon kosher salt, and 1/2 teaspoon black pepper into the toasted breadcrumbs until evenly moistened; set the filling aside.
Butterfly the flank steak to create one large, even sheet. Cut into 4 rectangles. Place each between sheets of plastic wrap and pound to about 1/4-inch thickness.
Lay prosciutto over each piece of beef. Divide the breadcrumb mixture among them, pressing lightly. Roll from the short end into tight logs and tie with kitchen twine every 1 to 1 1/2 inches. Lightly season the outsides with salt and pepper.
Heat 1 tablespoon olive oil in a heavy Dutch oven over medium-high. Sear the rolls on all sides until well browned, about 2–3 minutes per side. Work in batches if needed. Transfer the seared braciole to a plate.
Reduce heat to medium and add 1 tablespoon olive oil if the pot looks dry. Add the sliced garlic and cook just until fragrant, 30–60 seconds. Pour in the red wine and simmer 2 minutes, scraping up any browned bits.
Stir in crushed tomatoes, 1 teaspoon kosher salt, a few grinds of black pepper, and the red pepper flakes. Bring to a gentle simmer. Nestle the braciole into the sauce and spoon some sauce over the tops. Cover and transfer to the oven.
Braise until the beef is very tender, about 2 hours 15 minutes, turning the rolls once halfway through. Remove from the oven and let rest, covered, for 10 minutes.
Cut away the twine and slice the braciole into 1/2-inch rounds. Return the slices to the pot and warm gently on low heat for 5 minutes so they absorb more sauce. Off the heat, scatter torn basil over the top.
Serve with extra Parmesan over cooked pasta or creamy polenta, spooning plenty of sauce over each portion.