Heat the olive oil and butter together in a medium saucepan over medium heat until the butter is melted.
Add the finely chopped onion to the saucepan and sauté for 5 minutes until the onion is soft and translucent.
Add the minced garlic and cook for an additional 1 minute, stirring frequently to prevent burning.
Sprinkle the all-purpose flour over the onion and garlic mixture, stirring constantly to form a roux. Cook the roux for 2 to 3 minutes until it turns a light golden brown color.
Gradually whisk in the beef broth and chicken broth, ensuring no lumps form.
Add the Worcestershire sauce, dried thyme, black pepper, and salt to the gravy mixture. Stir well to combine.
Bring the gravy to a gentle boil, then reduce heat to low and simmer for 15 minutes, stirring occasionally, until the gravy thickens to your desired consistency.
Taste and adjust seasoning with additional salt and pepper if needed.
Remove from heat and serve warm over your favorite dishes.