Top each serving with 1 tablespoon chopped fresh basil or Italian parsley, if desired.
Mix the tomatoes with their juices and tomato paste in a slow cooker until blended. Stir in the peppers, mushrooms, and onions.
Cut the meat into 8 pieces; pound to ½-inch thick. Sprinkle with the oregano. Add to the slow cooker; press into the sauce with the back of a spoon. Cover with the lid; cook on low for 4 to 4 ½ hours or on high for 2½ to 3 hours.
Cook the pasta as directed on package, omitting salt. Transfer the meat to a plate, reserving the sauce in the slow cooker; cover the meat to keep warm. Mix the cornstarch and water until blended. Add to the sauce; stir. Cook, covered, on high for 6 to 8 minutes or until thickened.
Drain the pasta; spoon onto serving plates. Top with the meat, sauce, and cheeses.