While preheating oven to 400°F/200°C/Gas Mark 6, grease several cookie sheets.
In a large bowl, mix both flours, oats and cornmeal. Make an indentation in the center of the dry ingredient pile and add the liquid ingredients gradually: water, oil, eggs, peanut butter and vanilla extract. Knead well with your hands.
Dust the work surface; roll the dough out to ½ inch thick. Cut out shapes with dog biscuit cookie cutters. Transfer biscuits to prepared sheets.
Bake for 20 minutes, then turn the oven off and let the cookies sit inside for another 20 minutes – over this time they will harden.