In a small pot, bring the water, sweet potato, cinnamon stick and aniseed to a boil. Simmer until the potatoes are tender, about 15 minutes. Set 1/2 cup (125 ml) of the cooking liquid aside. Drain the potatoes and return to the pot. Discard the cinnamon stick. Mash the potatoes with a potato masher until smooth, then transfer to a bowl and let cool for 5 minutes.
Add the sugar, yeast and salt to the potato purée. Mix well with a wooden spoon. Add the flour alternately with the reserved cooking liquid and mix until the dough is uniform. Cover with plastic wrap and let rise in a warm spot until doubled in volume, about 1 hour.
Preheat the oil in a deep fryer to 350°F (180°C). Line a baking sheet with paper towels. With moistened hands, make the doughnuts by shaping about 2 tbsp (30 ml) of dough into a ring and carefully dropping it into the hot oil. Watch out for splattering. Fry 2 or 3 doughnuts at a time until golden brown, about 2 minutes per side. Drain on the paper towels.
In a small pot, bring all the ingredients to a boil. Continue boiling until the liquid is the consistency of syrup, about 2 or 3 minutes. Drizzle over the doughnuts and serve immediately.
Recipes with the "Healthy Pick" stamp have been evaluated by a registered dietitian member of the Ordre professionnel des diététistes du Québec.