Mix beef, breadcrumbs, egg, ketchup, patty Worcestershire, 1/2 teaspoon salt and pepper lightly. Form 4 oval patties about 1/2 inch thick.
Heat oil in a deep 12-inch 5-quart saute pan over medium-high heat. Brown patties for 2 minutes per side and transfer to a plate while partly cooked.
Cook onion in the same pan for 8 to 10 minutes until soft and golden. Add broth, gravy Worcestershire, remaining salt and rotini and scrape the bottom.
Bring to a simmer and return patties on top of the pasta. Cover loosely and cook over medium-low heat for 12 to 15 minutes, gently stirring pasta around the patties every 3 minutes and adding reserved water if needed, until pasta is tender and every patty reaches 160°F.
Transfer patties to a clean plate. Mix cornstarch with cold water and stir into the simmering pasta for 1 to 2 minutes until the gravy lightly thickens.
Return patties to the pan, spoon gravy over them and serve one patty with a quarter of the onion pasta per person.