Combine the ground beef, grated onion, breadcrumbs, beaten egg, minced garlic, salt, pepper, Italian seasoning, and Worcestershire sauce in a large mixing bowl. Mix well until all the ingredients are evenly distributed.
Divide the mixture into 6 equal portions and shape each portion into a 3/4-inch thick patty.
Heat 2 tablespoons olive oil in a large skillet set over medium-high heat. Once the oil is hot, add the patties and cook for 4 to 5 minutes per side or until golden brown and cooked through. Remove the patties from the skillet and set aside.
In the same skillet, add 1 tablespoon of olive oil and 2 tablespoons of butter. Once the butter has melted, add the chopped onion and sliced mushrooms. Cook for 5 to 6 minutes or until the onions are translucent and the mushrooms have released their moisture.
Add the minced garlic and fresh thyme leaves to the skillet and cook for an additional minute or until fragrant.
Sprinkle the all-purpose flour over the mushroom mixture and stir to combine. Cook 1 minute or until the flour has been absorbed.
Slowly pour in the beef broth and Worcestershire sauce, constantly stirring to prevent lumps from forming. Bring the mixture to a boil, then reduce the heat to medium-low and simmer for 5 to 7 minutes or until the gravy has thickened.
Season the gravy with salt and pepper.
Return the cooked patties to the skillet and spoon the mushroom gravy over them. Cover the skillet and let the patties simmer in the gravy for 5 minutes or until heated through.
Garnish with chopped parsley and serve hot.