Preparation

Roast the vegetables

• Position a rack in the top third of the oven and heat the oven to 450°F.

  In a large bowl, toss the mushrooms, onions, and Brussels sprouts with 2 Tbs. of the oil, the garlic, thyme, 1-1/2 tsp. salt, and a few grinds of pepper. Spread on a large rimmed baking sheet. Roast for 20 minutes, stir the vegetables, and continue roasting until tender and browned, about 35 minutes total. Transfer to a serving bowl and toss with the remaining 2 Tbs. oil and the vinegar.

Make the millet polenta

• Meanwhile, rinse and drain the millet. Put it in a heavy-duty 4-quart saucepan and stir over medium-high heat until it smells toasty and turns deeply golden, 7 to 8 minutes.

  Add 5 cups of the vegetable broth and the butter and bring to a boil over medium-high heat. Reduce the heat to a simmer and cover. Allow the millet to simmer gently, stirring after the first 20 minutes and then every 7 to 8 minutes thereafter to prevent sticking, until it becomes a thick, creamy porridge with a chewy texture, about 35 minutes total. If it seems too thick, stir in a little more broth. Stir in the cheese and season to taste with salt, if needed.

  Serve the polenta in wide, shallow bowls with the roasted vegetables on top. Drizzle a little olive oil or melted butter over each serving, if you like.