Preheat the oven to 400 degrees F (200 degrees C). Line a baking sheet with parchment paper.

Place 4 cups oatmeal in the bowl of a food processor; pulse until finely ground. Transfer to a mixing bowl.

Stir in grated apple. Mix in canned pumpkin and egg until well combined; mixture will be thick and slightly sticky.

Roll dough on a surface sprinkled with remaining 1/4 cup oats to a 1/2-inch thickness. Cut shapes with a cookie cutter and transfer to the prepared baking sheet.

Bake in the preheated oven until crisp and golden, 12 to 15 minutes. Remove to a wire rack to cool completely.