In a medium saucepan over medium-high heat, combine 3 cups of the milk, the rice, and brown sugar. Cook, stirring, until the mixture begins to boil. Reduce the heat to medium-low and continue cooking, stirring frequently, until the rice is tender and most of the milk is absorbed. This will take about 25 to 30 minutes.
Meanwhile, combine the pumpkin with the vanilla, cinnamon, and salt. When the rice is done, remove from the heat and stir in the pumpkin mixture. Cover and refrigerate until serving time.
To warm the pudding, stir in about ½ cup of milk or half-and-half and warm over medium-low heat, stirring constantly. Serve with a sprinkling of cinnamon and brown sugar, or top with a dollop of whipped cream or whipped topping and chopped walnuts.