Preheat oven to 350 degrees F. Grease a 1-1/2-quart casserole; set aside. Prepare vermicelli according to package directions (including the standing time).
Meanwhile, in a large skillet, cook ground meat over medium heat until brown. Drain off fat. Stir in carrot, cream cheese, and milk. Stir in cooked vermicelli and, if desired, a few basil leaves. Transfer mixture to the prepared casserole. If desired, toss bread crumbs with melted butter and sprinkle over casserole.
Bake, uncovered, for 25 to 30 minutes or until heated through, stirring once halfway through baking. Let stand for 5 minutes before serving. Makes 4 servings.