Preheat oven to 400 degrees F. Line 2 cookie sheets with parchment paper; set aside.
In a medium bowl, whisk together the whole wheat flour, old-fashioned oats, and cornmeal. Set aside. Meanwhile, in a large bowl, combine the water, oil, eggs, and peanut butter and whisk together until combined and smooth. Add in the flour and stir until a thick ball of dough forms.
Dust a clean, flat work surface with cornmeal and roll the dough out to about 1/4-1/2" thickness, depending on how thick you'd like them. Using a small cookie cutter (the dog one I used was about 1.5" long), cut out shapes and place 1-2" apart on the prepared baking sheets. Reroll and cut out more shapes.
Bake for 20 minutes, rotating pans halfway through baking time to ensure even baking. When you remove the cookies from the oven, the sheet may look slick with oil - this is okay and will not make the cookies soggy or oily. Allow the cookies to cool on the baking sheets completely until room temperature and set before serving. Store airtight at room temperature up to one week.