In a mixing bowl, combine the ground beef, Italian bread crumbs, rosemary, and 1 teaspoon of Worcestershire sauce. Divide the mixture into four equal portions and shape them into oval patties.
In a large skillet, heat 2 tablespoons of olive oil and 1 teaspoon of butter over medium-high heat. Gently place the patties in the skillet, ensuring not to overcrowd them.
Fry the patties for approximately 4 to 5 minutes on each side. You’ll know they’re ready to flip when they develop a beautiful golden-brown crust.
In the same skillet, add the onions. Cook them over medium heat until they are soft and lightly browned, about 5 minutes.
Once the onions are tender, pour in the beef broth and add the ketchup, remaining Worcestershire sauce, and season with salt and pepper to taste.
In a small bowl, whisk together the cornstarch and 1 tablespoon of the gravy mixture to create a thin paste. Slowly whisk this paste back into the gravy mixture, stirring continuously.
Bring the mixture back to a gentle boil, then reduce the heat to low. Let it simmer for a couple of minutes until the gravy thickens significantly.
Add the browned patties back to the skillet, ensuring they are coated with the gravy. Let them heat through for another 2 to 3 minutes.
Once everything is heated, taste the gravy and adjust the seasoning if necessary.
Serve the Hawaiian Hamburger Steak on a plate, generously topped with the rich gravy.