Heat 1 tablespoon of oil and 1 tablespoon butter in a heavy large skillet over medium-high heat. Add the carrot, onion, and garlic and saute until the carrots are tender, about 10 minutes. Season with salt and pepper. Remove from the heat and cool slightly. Stir in the Parmesan. Season beef with salt and pepper. Using 2 tablespoons for each, divide the carrot mixture among the beef slices. Roll up the beef slices to enclose the filling, tucking in the ends. Secure the rolls with toothpicks.
Heat the remaining oil and the remaining tablespoon of butter in the same large frying pan over medium-high heat. Dredge the beef rolls in flour to coat lightly. Add the beef rolls and cook until brown on all sides, about 8 minutes. Add the wine and stir to scrape up any browned bits on the bottom of the skillet. Add the tomatoes with their juices and the bay leaf. Cover and simmer until the beef is cooked through, about 5 minutes. Transfer the beef rolls to a platter. Remove the toothpicks, then cover the beef rolls with foil to keep warm. Add the peas and prosciutto to the sauce in the skillet and stir until the peas are heated through and the sauce reduces slightly, about 5 minutes. Season the sauce, to taste, with salt and pepper. Drizzle the sauce over the beef rolls and serve.