Place the sourdough starter in a large bowl or the bowl of a stand mixer along with the whole wheat flour, the bread flour, sugar, salt, non-dairy milk and butter. Mix, trickling in water as needed until a non-sticky dough forms. Continue kneading another 3-4 minutes by hand or in the stand mixer until the dough gets smooth and supple. It should not be too firm.
Form the dough into a smooth ball and place in an oiled bowl, turning over once to coat the top with the oil. Cover tightly and set aside in a warm place for eight hours or until risen and puffy. It may not double.
Flour your work surface. Remove the dough from the bowl and flatten into a disc. Using a rolling pin, roll the dough to an even, 1/2-inch thickness. If the dough is too resistant to rolling, let it rest 10 minutes, covered, then roll.
Use a 3-inch cookie cutter to cut out discs for the muffins. Roll up any scraps, knead briefly, and cut out more. You should get 12 muffins.
Sprinkle a baking sheet with cornmeal or semolina. Place the muffins on it, at least an inch apart. Sprinkle some more cornmeal on the muffins, cover with a kitchen towel, and let them stand at least 3-4 hours in a warm place. Again, they'll look puffier but may not double or rise too much.
To bake the muffins, heat a cast iron or any nonstick griddle on medium heat. Turn the heat to medium-low and place the muffins in the griddle without overcrowding, at least an inch apart.
Cover the griddle, set your timer for five minutes, and let the muffins cook. You might want to check at the end of three minutes to make sure they are not overbrowning. If they are, turn the heat down.
Once the underside is golden-brown, flip over and cook another four minutes, covered.
Let the muffins cool on a rack before you cut into half, toast and serve.