|Combine chicken broth, potatoes, rutabaga, parsnips, garlic, bay leaf, and thyme in a large pot. Bring to a boil. Reduce heat, and cover partially. Simmer until vegetables are very tender, about 30 minutes. Drain.
Transfer vegetables to large bowl. Add 1/2 cup butter. Using an electric mixer, beat mixture until mashed but still chunky. Season with salt and pepper. Transfer mashed vegetables to a buttered 13 x 9 x 2 inch baking dish.
Melt remaining 1/4 cup butter in a large, heavy skillet over medium-high heat. Add sliced onions. Saute until they begin to brown, about 5 minutes. Reduce heat to medium-low. Saute until onions are tender and golden brown, about 15 minutes. Season with salt and pepper.
Spread onions evenly over mashed vegetables. At this point, you can bake, cover and refrigerate. Prepared casserole will keep for up to a day in the fridge.
Preheat oven to 375 degrees. Bake, uncovered, for 25 minutes, or until heated through and top begins to crisp.