Serves 6 -8
1/4 cup extra-virgin olive oil
6 cups thinly sliced sweet onions (from 2 large)
1 medium clove garlic, thinly sliced
1 tablespoon plus 1/2 teaspoon cider vinegar; more as needed
1 teaspoon chopped fresh thyme leaves
Freshly ground black pepper
1-1/4 pound Brussels sprouts, trimmed and halved or quartered if large
Position a rack in the center of the oven and heat the oven to 450°F.
Heat 2 Tbs. of the olive oil in a 5-quart heavy-duty pot over medium-high heat. Add the onions and garlic, reduce the heat to medium, and cook without stirring until the onions begin to brown on the bottom, 5 to 7 minutes. Sprinkle with 1/4 tsp. salt and stir with a wooden spatula. Continue to cook, stirring and scraping the bottom of the pot frequently and adjusting the heat as necessary, until the onions are very soft and evenly browned, 20 to 25 minutes. Stir in the vinegar and cook for 1 minute. Stir in the thyme and season to taste with salt and pepper.
Meanwhile, toss the sprouts with the remaining 2 tablespoons oil and season generously with salt and pepper. Roast in a single layer on a large rimmed baking sheet, stirring once or twice, until tender, 20 to 25 minutes.
Toss the Brussels sprouts with the onions. Season to taste with salt, pepper, and vinegar, and serve.