Make the shallots. Preheat oven to 200ºC/Fan 180ºC/Gas 6.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. Remove from oven and leave to cool.
While the shallots are roasting, make the celeriac puree. 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 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.
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.