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Preheat the oven to 425°F and set an oven rack in the middle position. Line a 13 x 18-inch baking sheet with heavy-duty aluminum foil for easy clean-up, if desired.
Directly on the baking sheet, toss the onion, carrots, and Brussels sprouts, oil, salt, pepper, and thyme until the vegetables are evenly coated. Place the sausages on the baking sheet, spacing them evenly around the pan, and arrange the vegetables as best you can in a single layer. Roast for 25 minutes, until the sausages are cooked through and the vegetables are caramelized and tender (no need to stir while cooking).
Note: Feel free to substitute different vegetables, such as baby potatoes (halved), sweet potatoes (cut into 1-inch chunks), large broccoli or cauliflower florets, or even grapes. (If using potatoes, do not line the pan with foil, as they are prone to sticking.)