Heat oil in a medium saucepan over medium heat. Add onion and garlic; cook, stirring often, until softened, about 5 minutes. Add mushrooms and cook, stirring often, until they begin to release their juices, about 10 minutes.
Add broth, tamari (or soy sauce), thyme and sage; simmer for 10 minutes. Mix cornstarch and water in a small bowl. Stir into the sauce and simmer, stirring often, until slightly thickened, about 10 minutes more. Season with pepper. If you prefer a smooth gravy, pass it through a fine sieve (discard mushrooms and onions). Serve hot.