Place onions cut side down on baking sheet. Drizzle with 2 tablespoons oil and sprinkle with sugar. Bake for 30 minutes. Turn onions over and bake until brown and caramelized, about 30 minutes longer. Set aside to cool.
Place onions in food processor, add remaining oil, broth and vinegar. Puree until smooth and thick. Season with salt and pepper. Cover and refrigerate until chilled.
In a large salad bowl, combine greens, 1/2 cup walnuts and half of red onion. Add enough dressing to coat greens, toss well. Sprinkle with remaining walnuts and red onion and serve.