Melt 2 tablespoons butter in a Dutch oven or large saucepan with a lid over medium heat. Add the shallots and cook until just beginning to turn golden about 5 minutes. Add the carrots and celery root and season with salt and pepper, to taste. Stir in the stock, and enough water to just cover the vegetables. Bring to a boil, then reduce the heat to a simmer, cover, and cook until vegetables are fork tender, about 30 minutes.
Drain the vegetables in a colander set over a bowl to reserve the cooking liquid. Transfer the vegetables to a food processor and season with salt and pepper, to taste. Puree until very smooth, adding cooking liquid, as needed, to reach the desired consistency. Taste and season again, if necessary. Transfer to a serving bowl and serve warm.
Cook's Note: This can be made the day before and slowly reheated.