Bring water to a boil in a saucepan; add oats, pumpkin puree, vanilla extract, 1/4 teaspoon, cinnamon, and ginger. Reduce heat to low and simmer, without stirring, until oatmeal is thickened, about 25 minutes.
Pour 3/4 cup milk into oatmeal, stir well, and cook for 10 minutes more. Spoon oatmeal into serving bowls and top with remaining 5 tablespoons milk, brown sugar, and 1/4 teaspoon cinnamon.
