Preheat the oven to 450 degrees F. Wash and trim the vegetables and cut the biggest beets in half. Spread the vegetables out on a rimmed baking sheet. Drizzle with olive oil and sprinkle with salt and pepper, liberally. Shake the pan to coat.
Roast the veggies in the oven for 20 to 30 minutes, until tender on the inside, but crisp on the outside.
Meanwhile, place the butter in a small skillet and place the skillet over medium heat. Once the butter melts, add the garlic and saute for 1-2 minutes. Then pour in the mirin and water, and whisk in the miso paste. Bring to a boil, and simmer for 1-2 minutes. Then turn off the heat.
When ready to serve, place the hot crispy vegetables on a platter and drizzle the miso glaze over the top. Then sprinkle with green onions.