Preheat oven to 425ºF.
Toss the quartered shallots with the olive oil, balsamic vinegar and thyme in a small baking dish. Season with salt and pepper. Bake for 40-50 minutes, stirring every 15 minutes or so, until the shallots are tender and beginning to brown. Set aside.
While the shallots are roasting, start the puree. Combine the water, milk and salt in a large saucepan set over high heat. Bring to a boil, then add the celery root and potato; return to a boil. Reduce heat to medium and simmer until the vegetables are tender, about 30 minutes. Drain.
Combine the celery root, potatoes and butter in a food processor and puree until smooth. Season to taste with salt and white pepper.
Transfer the celery root puree to a serving bowl and form a well in the middle; spoon the balsamic roasted shallots into the well and serve.