Preheat the oven to 350℉. Line two large baking sheets with unbleached parchment paper.
In a large mixing bowl, combine the gluten-free flour, oat flour, sugar, baking soda, and salt. Stir to combine.
Add the egg, melted butter (or coconut oil), maple syrup, and vanilla extract. Begin to stir everything together. If the dough seems too dry, add 1-2 T. of warm water to the dough. Stir until a smooth dough forms. As the dough comes together, you may want to use your hands to knead it into a smooth ball.
The dough should be soft and pliable. Not sticky, but not crumbly. Once you have a smooth dough formed, transfer the dough onto a lightly floured piece of parchment paper.
Place another piece of parchment paper over the dough and roll the dough out into a sheet that is about 1/4" thick. Use your animal cookie cutters to cut animal cookies out of the dough and transfer the cookies onto the baking sheets. You can place them fairly close together since they will only puff up a little.
Continue to re-roll out the dough and cut out animal cookies until all the dough has been used up.
Bake the animal cookies for 10-12 minutes. The bottoms of the cookies should be lightly browned. Allow them to cool for 5 minutes on the pans before transferring them to a cooling rack to cool completely.
Store the completely cooled animal cookies in an airtight container. They can be stored at room temperature for up to 4 weeks.