Combine the sweet potato, squash, 2 cinnamon sticks, 1/2 teaspoon anise seeds and 2 cups water in a saucepan. Bring to a boil over medium-high heat, then reduce the heat to medium low, cover and cook until the vegetables are tender, about 15 minutes. Remove the vegetables with a slotted spoon, transfer to a bowl and mash with a fork. Discard the cinnamon sticks; let the liquid cool until just warm.
Transfer 1/4 cup of the warm cooking liquid to a small bowl and sprinkle the yeast on top. Set aside until foamy, about 8 minutes. Stir the salt, yeast mixture and another 1/2 cup cooking liquid into the potato mixture. Add the flour and knead by hand until combined (the dough will be sticky). Cover with plastic wrap; set aside in a warm place until doubled in size, 1 hour, 30 minutes to 2 hours.
Meanwhile, make the syrup: Bring the sugar, molasses, the remaining cinnamon stick and 1/2 teaspoon anise seeds, the orange zest, lemon juice and 1/2 cup water to a boil in a saucepan over medium-high heat. Reduce the heat to medium low and simmer until syrupy, 15 to 20 minutes. Discard the cinnamon stick and orange zest. Let the syrup cool slightly.
Heat 2 inches of vegetable oil in a deep skillet over medium-high heat until a deep-fry thermometer registers 360 degrees F. Fry heaping tablespoonfuls of the dough, turning once or twice, until golden brown, 3 to 4 minutes; drain on paper towels. Drizzle with the syrup.