Preheat oven to 350 degrees F (175 degrees C). Grease and 8x8-inch baking pan.
Stir oats, brown sugar, baking powder, cinnamon, nutmeg, pumpkin pie spice, and kosher salt together in a bowl; make a well in the center. Pour half-and-half, egg, and vanilla extract into well in the center of the oat mixture; stir to combine. Add pumpkin to half-and-half oat mixture and mix thoroughly. Pour pumpkin batter into prepared pan.
Bake in the preheated oven until a toothpick inserted into the center comes out clean, 25 to 30 minutes.
