Lightly grease a 9x13-inch baking dish.
In a large skillet over medium-high heat, add the ground round and cook, while crumbling, until the beef is no longer pink, about 5-7 minutes.
Drain the excess grease from the skillet.
With the stovetop on medium-high heat, add the tomatoes with basil, garlic, and oregano and their juices, the tomatoes with green chilies and their juices, the tomato paste, the salt, the Italian seasoning, and the black pepper and bring the mixture to a boil, while stirring frequently.
Reduce the stovetop heat to medium-low.
Simmer the beef mixture, uncovered, until the flavors start to meld together, about 5 minutes.
Transfer the skillet from the heat to a wire rack and set it aside.
Preheat the oven to 350 degrees F.
In a pot of salted, boiling water, cook the egg noodles to al dente, according to the package directions.
In a large bowl, add the cooked noodles, the chopped green onions, the sour cream, and the cream cheese and stir until blended.
Evenly add the noodle mixture into the prepared baking dish.
Evenly add the beef mixture on top of the egg noodle mixture.
Sprinkle the beef mixture with the cheddar cheese, the parmesan cheese, and the mozzarella cheese.
Bake, covered, for 35 minutes.
Uncover the baking dish and bake until the cheese is melted, about 5 minutes.
Transfer the bake from the oven to a wire rack and let it stand before serving, about 10-15 minutes.
Serve.