If baking, preheat the oven to 350°F (175°C) and lightly grease a donut pan with oil. If frying, heat at least 3 inches of oil in a Dutch oven or countertop deep fryer to 350°F (175°C), using a frying thermometer to maintain an accurate temperature.
In a large mixing bowl, combine the flour, sugar, baking powder, baking soda, matcha powder, nutmeg (if using), and salt. In a separate bowl or measuring jug, whisk together the plant-based milk, apple cider vinegar, vanilla extract, and oil until well blended. Pour the wet mixture into the dry ingredients and stir until the batter is smooth and no lumps remain.
For baked donuts: Spoon the batter evenly into the prepared donut pan, filling each cavity no more than ¾ full to allow room for expansion. Bake in the preheated oven for 10-12 minutes, or until a toothpick inserted into the center comes out clean. Let the donuts cool in the pan for 5 minutes before transferring them to a wire rack to cool completely.
For fried donuts: Use a frying thermometer to make sure the oil in your Dutch oven or countertop deep fryer has reached 350°F (175°C). Fill a donut dropper with the batter and drop a test donut or two back into the bowl of batter to make sure it it's dropping uniformly.
Working in batches of 4-5 donuts at a time, as not to overcrowd your fryer, drop the batter carefully into the hot oil. Fry each donut for 2-3 minutes on one side, then flip and fry for an additional 2-3 minutes, or until golden brown and cooked through. Transfer the donuts to wire cooling rack thats been placed over a baking pan to drain any excess oil.