In a heavy saucepan over high heat, combine wine, chicken broth, and beef broth. Bring to a boil; reduce heat and simmer until mixture is reduced to 2 cups, about an hour. (This can be prepared a day ahead — cover and refrigerate).

In a small bowl, mix butter and flour. Heat 1 tablespoon olive oil in a large skillet over medium heat. Sprinkle steaks with salt and pepper; cook in hot oil until medium-rare, about 4 minutes per side. Transfer steaks to a plate.

Stir shallots, garlic, and thyme into the skillet. Add 2 cups reduced wine mixture; bring to a boil, scraping up any browned bits. Whisk in butter mixture until smooth. Boil sauce until thick enough to coat a spoon, about 2 to 3 minutes.