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Adjust rack to top position and preheat oven to 425 degrees. Bring a large pot of salted water to a boil. Wash and dry all produce.
Peel and mince or grate garlic. Finely chop chili. Trim and thinly slice scallions, separating whites from greens; finely chop whites. Peel and mince ginger until you have 1 TBSP (2 TBSP for 4 servings). Trim and thinly slice mushrooms. Trim, peel, and halve carrot lengthwise; thinly slice crosswise into half-moons.