Heat the olive oil and butter in a medium saucepan over medium heat until melted and hot.
Add the finely chopped onion and cook, stirring occasionally, until softened and translucent, about 5 minutes.
Add the minced garlic and cook for 1 minute until fragrant.
Sprinkle the all-purpose flour over the onion and garlic mixture and stir continuously for 2 minutes to create a roux, ensuring the flour is fully incorporated and lightly browned.
Gradually whisk in the beef broth and chicken broth, stirring constantly to prevent lumps.
Add the Worcestershire sauce, dried thyme, black pepper, and salt. Stir to combine.
Bring the mixture to a simmer and cook, stirring occasionally, until the gravy thickens to your desired consistency, about 15-20 minutes.
Taste and adjust seasoning if necessary.
Remove from heat and serve warm over your favorite dishes.