Pound the beef cutlets 1/4-inch thin. Add the cheese, parsley and garlic to a small bowl and toss. Divide evenly over each cutlet. Roll up each cutlet and tie with butcher's twine to secure and to make a roll. Sprinkle each roll with salt and pepper.
Heat 2 tablespoons olive oil in a large, heavy-bottomed Dutch oven over medium-high heat. Add the beef rolls and the cubed beef and brown on all sides. Remove to a plate once browned. Add a touch more olive oil to the pan and add the carrots and onions and saute until tender. Add the wine and simmer for a minute or so, and then scrape up the browned bits on the bottom of the pan. Stir in the passata and bay leaves. Bring to a simmer, add the beef back into the pot and cover with a lid. Cook on low until thick and the meat begins to fall apart, flipping the rolls occasionally, 2 1/2 to 3 hours. (If too dry, add a half cup of water; if too loose, remove the lid to finish thickening).
Bring a large pot of salted water to a boil. Add the orecchiette, cook until al dente and drain.
Remove the bay leaves and discard from the sauce. Carefully remove the braciole to a cutting board and cut off the butcher's twine. Slice the beef rolls into rounds. Add the pasta to the sauce and toss all together. Serve on a large platter and top with the braciole rounds and sprinkle with lots of pecorino cheese.
Use canned crushed tomatoes if you cannot find passata di pomodoro.