Preheat the oven to 350 degrees F.
In a large mixing bowl, whisk together the flour, sugar, baking powder, salt, and nutmeg.
In a small saucepan on low heat, add in the non-dairy milk, apple cider vinegar, vanilla, egg replacer, and vegan butter. Whisk until the vegan butter melts and everything is smooth. Remove from heat and pour into the flour mixture. Stir together until all of the ingredients are thoroughly combined.
Place the donut mixture into a piping bag with a #4 tip, spray a donut pan with non-stick cooking spray, and pipe the donut mixture into the donut pan. Place into the oven and bake for 12 minutes, until the donuts begin to turn golden.
Remove the vegan vanilla donuts from the oven and allow cooling in the pan for 5 minutes before removing the donuts.
Serve plain or make a quick glaze by whisking together 1/2 cup powdered sugar, 1 tablespoon non-dairy milk, and ¼ teaspoon vanilla. Dip the donuts into the glaze, allow it to set, and serve.