In a large saucepan, combine the first 8 dry ingredients. Gradually add 1 cup of the half-and-half and mix well. Stir in the remaining half-and-half and slowly heat over medium heat until it comes barely to a boil. Remove from heat. Whisk egg substitute until lightly beaten. Very gradually whisk 1 cup of the hot mixture into the eggs. Then return all to the pan whisking constantly. Cook and stir over medium heat until mixture reaches 165 degrees.
Remove from heat and transfer to a metal bowl placed in an ice bath. Stir until cool. Add pumpkin, vanilla, and rum. Mix well.
Press plastic wrap onto surface of custard to prevent film from forming. Refrigerate for several hours or overnight.
Fill cylinder of 2 quart ice cream maker and follow manufacturer's instructions. Optional (although this will add traces of fat, but so worth it): Add 2/3 cup graham crackers crumbled into 1/4 inch morsels and 1/3 cup white chocolate chips the last 5 minutes of churning.
Place in container and allow to firm up in freezer 2-4 hours before serving.
Note: The nutritional info for fat-free half-and-half is incorrect and throwing off the total fat calculation count. It should calculate to almost zero total fat.