Preparation Combine first 7 ingredients in heavy large pot. Add enough water to cover. Sprinkle with salt. Bring to boil, reduce heat to medium, and simmer with lid slightly ajar until vegetables are tender, 15 to 20 minutes. Drain; return to pot. Discard thyme sprigs and bay leaf. Stir over medium heat to dry vegetables. Using potato masher, mash vegetables until coarsely pureed. Mash in 3 1/2 tablespoons butter. Season with salt and pepper. DO AHEAD: Can be made 1 day ahead. Transfer to microwave-safe bowl; cover and chill. Rewarm in microwave before serving. Preheat oven to 425°F. Cut Jerusalem artichokes into 1/2-inch cubes. Place in medium bowl; add oil, sprinkle with salt and pepper, and toss to coat. Dot with remaining 1/2 tablespoon butter. Transfer to rimmed baking sheet; roast until tender and golden brown, turning occasionally, about 25 minutes. Place celery root and potato puree in serving bowl. Sprinkle Jerusalem artichokes and chopped thyme over and serve.
Preparation Combine first 7 ingredients in heavy large pot. Add enough water to cover. Sprinkle with salt. Bring to boil, reduce heat to medium, and simmer with lid slightly ajar until vegetables are tender, 15 to 20 minutes. Drain; return to pot. Discard thyme sprigs and bay leaf. Stir over medium heat to dry vegetables. Using potato masher, mash vegetables until coarsely pureed. Mash in 3 1/2 tablespoons butter. Season with salt and pepper. DO AHEAD: Can be made 1 day ahead. Transfer to microwave-safe bowl; cover and chill. Rewarm in microwave before serving. Preheat oven to 425°F. Cut Jerusalem artichokes into 1/2-inch cubes. Place in medium bowl; add oil, sprinkle with salt and pepper, and toss to coat. Dot with remaining 1/2 tablespoon butter. Transfer to rimmed baking sheet; roast until tender and golden brown, turning occasionally, about 25 minutes. Place celery root and potato puree in serving bowl. Sprinkle Jerusalem artichokes and chopped thyme over and serve.