Preheat oven to 425°F. Spread broccoli, mushrooms, and red onion on a large rimmed baking sheet. Toss with oil, 1/2 tsp. of the salt, and black pepper. Bake for 23 to 27 minutes, tossing once halfway through.
Meanwhile, prepare quinoa by combining quinoa with 2 cups water and remaining 1/2 tsp salt in a saucepan. Bring mixture to a boil, reduce to a simmer, cover and cook until water is absorbed, about 15 minutes. Remove from heat and let steam, covered, 5 additional minutes; fluff with a fork.
Prepare sauce by whisking vinegar, sesame oil, miso, turmeric, and ginger in a small bowl until smooth.
Divide quinoa and veggies evenly into each of 3 or 4 bowls and spoon miso-turmeric dressing overtop. Garnish with cashews and green onion, if desired.