Preheat the oven to 450 degrees Fahrenheit/235 degrees Celsius.
In a bowl, whisk together all the wet ingredients - sourdough starter, vegan yogurt and non-dairy milk. Set aside.
Place the all-purpose flour and whole wheat flour in a large bowl along with the baking soda and salt. Whisk to mix.
Make a well in the center of the flour and pour in the sourdough mixture, in three batches. Use a wooden spoon or your hand to mix after each addition, but don't knead the dough - make a claw-like shape with your hand and use a circular motion to mix the wet ingredients into the dry until all of the dough is evenly moistened and holds together in a shaggy ball. If there is still dry flour in the bowl after you've added all the liquid mixture, add non-dairy milk, a teaspoon at a time, until the dough comes together.
Turn out the dough on a floured surface and shape it into a ball without kneading it. This doesn't have to be perfect - soda bread is a rustic bread and it needs to look rustic for authenticity.
Flatten the ball slightly into a disc, then place it in a cast iron pan. Use a knife to cut a cross in the bread, going almost all the way down, but not cutting through.
Bake in the preheated oven for 30 minutes. Remove the bread from the oven and let it cool on a rack for 15-20 minutes before you slice it.