Tip:Although a Dutch oven (a deep, straight-sided flameproof casserole with a tight-fitting lid) is traditional for this recipe, a straight-sided sauté pan works, too. If you use a sauté pan, it should be about 3 inches deep and no more than 9 inches in diameter. That way, the vegetables will be snugly packed. Be sure it has a lid. Be extra careful when you take a sauté pan out of the oven: The handle will be extremely hot, so tie a kitchen towel around it to prevent burning yourself or others who might try to touch the pan, not realizing that it's been in the oven.
Heat the oven to 350°F. Cut the carrots and parsnip on the diagonal into 1-inch chunks. Cut the turnips and onions into wedges about 1 inch thick at their widest point. Split the fennel bulb lengthwise, notch out the core, and then cut the fennel crosswise into 1-inch slices. Heat the olive oil in a medium (9- inch) Dutch oven over medium heat. Add the onions and cook, stirring occasionally, until softened and lightly browned, about 5 min. Add the other vegetables, the bouquet garni, 1/2 tsp. salt, and pepper to taste. Raise the heat to medium high and cook until the vegetables are lightly browned, 5 to 7 min. Add the water or broth and bring to a boil. Cover the pot, put it in the oven, and bake until the vegetables are fully cooked but still hold their shape, 20 to 25 min. With a slotted spoon, transfer the vegetables to a serving bowl. Bring the pan juices to a boil over medium heat.(If there's a lot of liquid left, boil until reduced to about 1/4 cup.) Discard the bouquet garni. Whisk in the butter and herbs. Spoon the sauce over the vegetables and serve.