Finely dice the onion. Add the onion, olive oil, and ground beef to a large deep skillet and sauté over medium heat until the beef is fully browned and the onion is soft and translucent. Drain off any excess fat, if needed.
Add the flour to the skillet and continue to stir and cook for one minute more. The flour will begin to coat the bottom of the skillet. Be careful not to let the flour burn.
Add the tomato sauce and beef broth to the skillet and stir to dissolve the flour off the bottom of the skillet.
Add the uncooked pasta to the skillet and stir to combine. The liquid in the skillet may not fully cover the pasta, but that is okay.
Place a lid on the skillet, turn the heat up to medium-high, and let the liquid come up to a boil. Once it reaches a boil, give it a quick stir to loosen any pasta from the bottom of the skillet, replace the lid, and turn the heat down to low. Let the pasta simmer, stirring occasionally (always replacing the lid), for about 10 minutes, or until the pasta is tender and the sauce has thickened. Turn the heat off.
Add the shredded cheddar to the skillet and stir until it has melted into the sauce. Stir the hot dog relish into the sauce and then top with sliced green onions. Serve hot.