Stir milk and sourdough starter together in a large bowl until mostly smooth. Mix in flour. Cover bowl and leave to ferment at room temperature, 8 hours to overnight.

Add sugar, baking soda, and salt to the preferment, stirring just enough to incorporate. Turn dough out onto a lightly floured work surface. Knead for 4 to 5 minutes, adding as much of the 1/2 cup flour as needed to make the dough workable.

Use a rolling pin to flatten dough to an even 1/2-inch thickness. Cut into 4-inch rounds using a clean tuna can or biscuit cutter; keep edges well-floured to avoid sticking.

Sprinkle a thin layer of cornmeal over a sheet of waxed paper. Arrange cut-outs over the cornmeal. Sprinkle a thin layer over the dough. Let rest for 45 minutes.