Bring milk to boil in a small pot over medium-high. Stir oats, brown sugar, white sugar, cinnamon, pumpkin pie spice, and salt into the boiling milk. Place a cover on the pot, reduce heat to medium-low, and cook, stirring occasionally, for 3 minutes. Stir pumpkin puree into oat mixture until smooth; cook and stir continually another 2 minutes.
Remove from heat; let oatmeal set 2 minutes before serving.