In a large skillet, heat the olive oil over medium heat. Add the sausage and cook until browned on all sides, about 8-10 minutes. Remove the sausage from the skillet and set aside.
In the same skillet, add the sliced onions and cook until they are soft and caramelized, about 10-12 minutes. Stir in the minced garlic and cook for an additional minute.
Sprinkle the flour over the onions and stir to combine. Gradually pour in the beef broth while stirring to avoid lumps. Add the Worcestershire sauce and dried thyme. Bring the mixture to a simmer and let it cook for about 5 minutes until it thickens slightly.
Return the sausage to the skillet, season with salt and pepper, and simmer for another 5 minutes to heat through.
Serve the sausage and onion gravy over mashed potatoes or with crusty bread. Garnish with fresh parsley.