Place sweet potato cubes and garlic in a saucepan and pour in chicken or vegetable broth to within a half inch of the top of the potatoes. Boil until very soft (the potatoes will sink down into the liquid as they cook), 12-15 minutes. Drain, reserving cooking liquid.
Return potatoes to saucepan and mash with potato masher. Slowly add some of the cooking liquid, stirring until you get the consistency of a loose puree. It need not be perfectly smooth. Stir in thyme and butter, adding more to taste if needed, and season with salt and white pepper to taste. Keep warm in a double boiler. Drizzle aged balsamic vinegar over each portion just before serving.