Heat oil in a medium pot over medium heat. Add onion, cut side down, garlic, and ginger. Cook, stirring occasionally, until garlic is golden, 3–4 minutes. Add 1 1/2 cups water, broth, star anise, and cinnamon; bring to a boil. Reduce heat; simmer until flavors meld, 7–8 minutes. Add mushrooms; simmer 2 minutes. Add scallion. Season with salt.
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