Preheat a large saute pan over medium heat. Add the onion and mushrooms. Stirring occasionally, cook until dry and softened, about 10 minutes.
Add the garlic and thyme to the pan, stir, and cook for about 30 seconds. Sprinkle the flour into the pan, stir and cook for about 1 minute or until lightly browned.
Add the broth and milk, and whisk to dissolve the flour. Return to a low simmer, and cook for about 5 minutes or until thickened. Add pepper to taste.
Stir in the tamari, and reduce heat. Taste and adjust seasoning as desired. Serve immediately, or allow gravy to cool, then refrigerate until ready to re-heat and serve.
