Preheat oven to 350°F (177°C). In a baking dish, arrange onions in a single layer. Drizzle with vegetable oil and season with salt and pepper. Add thyme sprigs and toss until onions are evenly coated. Roast, turning twice, until onions are soft and well browned, about 40 minutes. Keep warm.
In a small bowl, whisk together vinegar, garlic, and extra-virgin olive oil. Season with salt and pepper. Place cabbage and chicory in a large bowl and toss with just enough dressing to coat. Add warm onions and walnuts and toss again, adding more dressing if needed. Transfer to a serving bowl or platter and top with goat cheese. Serve.