In a pot over medium heat, warm the plant-based milk to approximately 110°F (43°C). If you have one, use a candy thermometer to check the temperature, otherwise, just go for the rough feel of warm (not scalding) bath water.
Pour the warm milk into a large mixing bowl. Add sugar and instant yeast to the milk. Stir gently to combine and let it sit for about 5 minutes until the yeast activates, indicated by a foamy appearance.
Add bread flour, melted vegan butter, salt, nutmeg, and vanilla to the yeast mixture. Mix with a spoon until the dough starts to come together. Then, knead the dough in the bowl with your hands until it forms a smooth, elastic ball, which should take about 10 minutes, or mix it in a stand mixer on medium speed for 5 minutes. If the dough is too sticky, add a bit more flour, just enough to make it manageable.
Cover the dough with a clean kitchen towel and let it rise in a warm place, ideally around 75°F to 80°F (24°C to 27°C), for 1 hour, or until it doubles in size. After the sweet dough has risen, gently punch it down to release air. Transfer the dough to a lightly floured surface. Divide it into 12 equal portions.
On a clean lightly floured, clean surface or large cutting board, roll each dough ball into a long, thin rope, about 12 inches (30cm) long. Fold each rope in half and twist the two pieces around each other to form a spiral, pressing the ends together to seal. Once formed, slightly flattening the twisted dough helps prevent it from unwinding as it proves and bakes.
Place the twists on a lightly floured baking pan, leaving space between the twists, and place them in a warm area to proof for 20 minutes. An oven turned off is a good place for that, or anywhere where they will be protected from drafty air.
In a deep, heavy-bottomed pot, heat the oil over medium heat to 355°F (180°C)). Ideally, get an accurate temperature by using a thermometer. Otherwise, test the oil's readiness by dropping a small piece of dough into it; it should sizzle and float to the top.
Carefully place a few dough twists into the hot oil, avoiding overcrowding. Fry them for 2-3 minutes on each side, or until golden brown and puffed up.
Remove the twists from the oil with a slotted spoon or tongs, and drain them on a wire rack suspended over a baking sheet to drip off excess oil.
For the cinnamon sugar mix, combine sugar and ground cinnamon in a shallow bowl. Roll the fried doughnuts in this mixture until fully coated.
Serve the doughnut twists while warm for maximum magic!