Preheat oven to 375 degrees F (190 degrees C). Spray 6 muffin cups with cooking spray.
Whisk all-purpose flour, whole wheat flour, sugar, salt, pumpkin pie spice, baking powder, baking soda, cinnamon, and nutmeg together in a large bowl until thoroughly combined. Stir pumpkin puree, milk, eggs, and maple syrup into the dry ingredients until batter is smooth; fold oats into batter. Scoop batter into prepared muffin cups, filling them to the top.
Bake in the preheated oven until a toothpick inserted into the center of a muffin comes out clean, 20 to 30 minutes; set aside to cool 5 to 10 minutes before serving.