2 medium celery roots (celeriac) peeled, cut into 1-inch cubes
1 pound potatoes, peeled, cut into 3/4-inch cubes
2 cups chicken broth
2 cups milk
3 cloves garlic, peeled
3 fresh thyme sprigs
1 fresh bay leaf
4 tablespoons (1/2 stick) unsalted butter, divided
8 ounces sunchokes, scrubbed
1/2 tablespoon extra-virgin olive oil
1 teaspoon chopped fresh thyme (for garnish)
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 and 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. This can be done ahead.
Preheat oven to 425°F. Cut sunchokes 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 with roasted sunchokes and chopped thyme.