Preheat oven to 375F and butter a 9×13 inch glass baking dish.
Combine the broth, potatoes, rutabagas, parsnips, garlic, and bay leaf in a 4 quart soup pot, and bring the broth to a boil. Reduce the heat, cover partially, and simmer until the vegetables are very tender, about 30 minutes. Drain well. Remove the bay leaf.
Transfer the vegetables to a large bowl and add 1/2 cup of the butter or margarine. Using an electric mixer, beat the vegetables until mashed but still chunky.
Season with salt, pepper, and thyme and transfer the vegetables to the buttered baking dish.
Melt the remaining 1/4 cup butter or margarine in a large, heavy nonstick skillet over medium-high heat.
Add the sliced onions and sauté until beginning to brown, about 5 minutes.
Reduce the heat to medium-low and cook until the onions are tender and golden brown, about 15 minutes. Season the onions with salt and pepper and spread evenly over the mashed vegetables.
Bake uncovered, until heated through and the top begins to crisp, about 25 minutes.