A quick stir-fry brown sauce: Saute shopped garlic in canola oil until just browning. Add minced ginger. A tsp of each will do. Add a healthy tbsp of soy sauce, cook down until just starting to thicken. Stretch with a cup of chicken broth/1 tsp corn starch (mixed). It's done when shiny and slightly syrupy.
Some of the different types of oriental brown gravy are Ginger gravy in which you add ginger or Mushroom Gravy with sliced mushrooms garlic and ginger. Other types are Oyster Gravy made with some vegetarian Oyster sauce or Sesame Gravy with Chinese Sesame oil. All are very good and its hard just to pick one.
Well, in my experience such an oriental gravy that was rather nice was known as Almond Gravy, It was usually served in association with another dish, but could be ordered to accompany most anything upon request. Went well with anything from plain white rice to deep fried prawns.
Soy sauce is the obvious choice, there are many different types of soy sauce and they are everywhere! Garlic sauce is also very good and is sometimes dark brown. So is mushroom sauce.
Oriental brown gravies are sauces made of water, or stock, soy sauce, and oyster sauce, all thickened with a slurry made of cornstarch, It may also be black bean sauce.
There are several types of dark sauce in Asian cuisine.
There are those based on soy-based Apple, based on fish and some plant-based. Most liquid and most full-bodied.
You must remember if the flavor was salty, sweet or sour to choose from, in a specialized store.