Sprinklewith salt. Bring to boil, reduce heat tomedium, and simmer with lid slightly ajaruntil vegetables are tender, 15 to 20 minutes.
Drain; return to pot. Discard thyme sprigsand bay leaf. Stir over medium heat to dryvegetables. Using potato masher, mashvegetables until coarsely pureed. Mash in3 1/2 tablespoons butter. Season with salt andpepper. DO AHEAD: Can be made 1 day ahead.
Transfer to microwave-safe bowl; cover andchill. Rewarm in microwave before serving.
Preheat oven to 425°F.
Cut Jerusalemartichokes into 1/2-inch cubes.
Place in mediumbowl; add oil, sprinkle with salt and pepper,and toss to coat. Dot with remaining 1/2tablespoon butter.
Transfer to rimmed bakingsheet; roast until tender and golden brown,turning occasionally, about 25 minutes.
Place celery root and potato puree inserving bowl.
Sprinkle Jerusalem artichokesand chopped thyme over and serve.