Prepare a large pot with cooking spray and place over medium heat; cook the beef stew meat in the pot until completely browned on all sides, 5 to 7 minutes. Add the onions; cook and stir until the onions begin to soften, about 5 minutes. Stir the water, ketchup, Worcestershire sauce, vinegar, brown sugar, paprika, salt, and mustard into the beef and onion mixture. Place a cover on the pot, reduce heat to low, and simmer the mixture until the beef is tender, about 2 1/2 hours.
Bring a large pot of lightly salted water to a rolling boil. Cook the egg noodles at a boil until cooked through yet firm to the bite, about 5 minutes; drain. Serve the goulash over the noodles.
