Preheat oven to 325 degrees.
Cut the top-round slices in half widthwise to have 12 equal pieces. Place the beef on a cutting board or other flat surface. Place a zipper-top bag over each beef slice and gently pound to 1/4 to 1/8-inch thickness.
Mince 4 cloves of garlic; slice the remaining 8 cloves.
In a small bowl, combine the minced garlic, parsley, cheese, breadcrumbs, salt, and black pepper to taste.
Place a prosciutto slice over each piece of beef, then sprinkle the filling evenly over the prosciutto (2-3 tablespoons per slice).
Roll the beef into a cylinder, tucking in the sides to hold in the filling as you roll. Secure with toothpicks.
Heat the olive oil in a Dutch oven or large pot over medium-high heat. Working in two batches of six each, place the beef rolls in the pot, seam side down first, which will help to seal it. Cook, turning the meat occasionally until each roll is nicely browned on all sides, adjusting the heat as necessary to prevent burning. Transfer to a plate and repeat with the remaining six rolls.
Add the wine to the Dutch oven. Bring to a boil, scraping up any browned bits. Let the wine boil for 2-3 minutes to reduce it and cook off some alcohol.
Add beef broth, tomatoes, sliced garlic, and Italian seasoning. Return the beef rolls to the pot and bring back to a simmer.
OVEN: Place a dampened piece of parchment paper (or aluminum foil) over the Dutch oven, then place the cover over the pot. Place in the oven and cook for 1 1/2-2 hours, turning the rolls after 1 hour. (My sweet spot for tenderness is usually around 1 hour and 45 minutes).STOVETOP: Cover and cook on low heat, occasionally turning until beef is tender and easily pierced with a fork, about 1 1/2 hours.SLOW COOKER: 3-4 hours on high, 5-6 hours on medium, or 7-8 hours on low. Prop the lid up towards the end to allow some of the steam to escape and the flavor to concentrate. Transfer the beef braciole to a platter and tent with foil to keep warm. Remove the toothpicks.
Place the pot back on the stove. Put the flour in a heat-proof bowl, then remove some of the hot cooking liquid. Stir until smooth to create a slurry. Slowly add the slurry to the liquid in the pot, bring to a slow simmer, and cook until thickened.
Season with salt and black pepper to taste. Add a pinch of sugar if needed. Pour the sauce over the beef rolls on the platter. Garnish with chopped fresh parsley. Serve over pasta, potatoes, or polenta with additional grated cheese if desired.