To Prepare the Pumpkin:

1. At least 24 hours before you make the phyllo rolls, open the can of pumpkin and transfer the contents to a zipper-lock freezer bag. Flatten the bag and seal it. Place in the freezer and freeze for at least 24 hours or until frozen solid.
2. The morning of the day you make the phyllo rolls, take the bag of pumpkin out of the freezer and thaw it out. You can speed this up by using your microwave thawing option or by placing the bag in a bowl of hot water. Line a sieve or colander with a piece of fine cheesecloth or thin cotton sheeting material. Transfer the thawed pumpkin into the sieve and let it drain over a bowl for about 1/2 an hour. This makes a very thick pumpkin puree and should measure 1 1/2 cups. Set aside.

To Make the Syrup:

1. Mix together the two syrups and the water in a small stainless-steel pot and bring it to a boil. Turn the heat down and simmer the mixture over medium heat for 5 minutes. Stir in the lemon juice. Pour the syrup into a heat-proof bowl and set aside to cool. When it is no longer hot, refrigerate it.

To Make the Filling:

1. Mix the drained pumpkin in a medium bowl with the brown sugar, nuts, cornmeal, spices and salt. Mix well and set aside.

To Assemble the Phyllo Rolls:

1. Melt the vegan butter in a small bowl and keep it warm (place the bowl inside of a larger bowl of very hot water).
2. Separate 7 full sheets of phyllo from your box of phyllo (you can repackage and refrigerate the remaining phyllo, to use in another recipe). Keep the sheets you are not working with on a clean kitchen towel covered with plastic wrap.
3.  Preheat the oven to 400°F.
4. One at a time, lay a sheet of phyllo on the clean, dry surface covered with a sheet of baking parchment. Brush the surface of the phyllo lightly with the melted vegan butter, using a soft brush. Spread about 1/3 cup of the Filling along the long side nearest to you. Carefully roll the sheet away from you until a long roll results. Cover the roll and repeat until all are done. Cut each roll into 4 equal-sized "logs".
5. Carefully place the "logs" seam side down on a 12 x 17-inch baking sheet lined with baking parchment cut to fit. Brush the tops of the “logs” with the remaining melted vegan butter. Bake the rolls for 30 minutes, or until golden and crispy.
6. While the rolls are still hot, carefully remove them from the baking sheet to shallow 8 x 14-inch baking pan. When you remove the rolls from the oven, leave the oven on at 400°F so that you can roast the apples.
7. Pour the cooled syrup over the hot pastries. Allow the pastries to cool and soak up the syrup before serving.

To Make the Roasted Apples:

1. Toss all of the ingredients for the Roasted Apples together in an 8x12-inch baking pan and roast them for about 20 minutes, stirring several times or until the apples are soft and slightly browned. Remove them from the oven and transfer them to a shallow bowl to cool. Set aside.
2. For each serving, place 4 rolls on one of seven small dessert plates. Divide the Roasted Apples evenly between the plates.
3. Immediately before serving, top the rolls with a little whipped topping of your choice, or place 1/2 cup of vegan vanilla "ice cream" alongside instead.