Preheat an oven to 400 degrees F (200 degrees C). Grease 12 muffin cups, or line with paper muffin liners. Whisk the milk, egg, butter, and vanilla extract together in a bowl.
Mix the flour, oats, brown sugar, baking powder, baking soda, pumpkin pie spice, and salt in a bowl. Make a well in the center and pour in the egg mixture. Stir until just combined. Gently fold in the blackberries. Divide the batter evenly into the muffin cups, and sprinkle with the turbinado sugar.
Bake in the preheated oven until a toothpick inserted into the center of a muffin comes out clean, about 20 minutes. Cool in the pans for 10 minutes before removing to cool completely on a wire rack.