Preheat oven to 200ºC.
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.
Combine the water, milk and salt in a large saucepan set over high heat. Bring to a boil, then add the celeriac; return to a boil. Reduce heat to medium and simmer until the celeriac is tender, about 30 minutes. Drain.
Combine the celeriac and butter in a food processor and puree until smooth. Season to taste with salt and black pepper.
Transfer the celeriac puree to a serving bowl and form a well in the middle; spoon the balsamic roasted shallots into the well and serve.