Create buttermilk: Stir together the milk and vinegar. Add the vegan butter and microwave for 1 minute, until the butter is melted. Set aside to cool to 105 to 110°F (40 to 43°C).
Make the dough: Add the flour, yeast, salt, and sugar in the bowl of a stand mixer fitted with a paddle attachment. Beat for a second or two to combine. Test the buttermilk to make sure it's in the right temperature range. If so, add it to the flour and mix on low speed for 1 minute, until the dough is combined. Switch to the dough hook attachment and turn the mixer up to medium-high speed for 5 minutes. The dough should be smooth and elastic. Add up to 2 tablespoons of additional flour if the dough is too sticky. If you don't have a stand mixer, knead the dough by hand for 5 to 6 minutes on a lightly floured surface.
Raising the dough: Transfer the dough ball into a greased bowl. Cover with a moistened kitchen towel and place in a warm place until it doubles in size, approximately 1 hour. See post notes for the best place to let the dough rise.
Make muffins: When the dough is ready, spread cornmeal across a baking pan. Punch down the dough, divide it into 14 pieces, and roll each piece into a ball. Flatten each ball until they're roughly 3 inches in diameter. Place the dough pieces in the prepared pan and sprinkle the top of each with more cornmeal. Cover and set aside for 20 to 30 minutes.
Cook Muffins: Heat a griddle on low heat. Add a light coating of oil or spray with vegetable cooking spray. Use a spatula to carefully transfer the dough to the heated griddle. Cook for 8 to 10 minutes until the crust is golden brown. Carefully turn muffins and cook on the other side for 8 to 10 minutes. Keep an eye on the muffins and make sure they're not getting overly browned. If so, transfer them to a baking sheet and bake in the oven at 300°F/150°C for up to 10 minutes.
Bake: This step is optional, but to make sure the muffins are cooked through, transfer them to a baking pan lined with parchment paper and bake at 300°F/150°C for 10 minutes.
Cool: Transfer the cooked muffins to a wire rack to cool completely.