Bring a large pot of salted water to a boil. Add the wide egg noodles and cook according to the package directions until just al dente. Drain well and set aside.

While the noodles cook, heat the olive oil in a large deep skillet or sauté pan over medium-high heat. Add the ground beef and cook, breaking it into small pieces with a spoon, until browned and no longer pink.

Add the finely chopped yellow onion to the skillet with the beef. Cook for 3 to 4 minutes, stirring often, until the onion softens and turns translucent.

Stir in the minced garlic, salt, black pepper, and dried Italian seasoning. Cook for 1 minute more, just until the garlic is fragrant. Reduce the heat to medium.

Push the beef mixture to one side of the skillet. Add the unsalted butter to the empty side of the pan and let it melt. Sprinkle the all-purpose flour over the melted butter and whisk or stir to form a smooth paste.

Slowly pour in the whole milk while stirring constantly, then add the beef broth. Stir everything together, scraping up any browned bits from the bottom of the skillet. Simmer for 3 to 5 minutes, stirring, until the sauce thickens slightly.

Reduce the heat to low and add 1 1/2 cups of the shredded sharp cheddar cheese to the skillet. Stir until the cheese is fully melted and the sauce is smooth and creamy.

Add the drained egg noodles to the skillet and gently fold them into the cheesy beef mixture until everything is evenly coated and combined.

Sprinkle the remaining 1/2 cup shredded sharp cheddar cheese and the grated Parmesan cheese evenly over the top of the beef and noodles mixture in the skillet.

Cover the skillet with a lid and cook over low heat for 5 to 7 minutes, or until the cheese on top is melted and the mixture is bubbling gently around the edges.

Remove the skillet from the heat and let the beef and noodles and cheese rest for 5 minutes to set slightly. Sprinkle the chopped fresh parsley over the top just before serving.