Place non-dairy milk and date sugar in a small saucepan and heat over low heat until lukewarm.
Remove from heat and whisk in yeast. Let sit.
Meanwhile, mix together dry ingredients and set aside.
Whisk pumpkin or applesauce and vanilla into the milk mixture and then pour entire liquid mixture into the dry ingredients.
Gently mix and knead just until it forms a smooth dough. It should be tacky, but not overly sticky. Add additional milk if too dry (not tacky) or flour if too sticky.
Roll dough out into a large rectangle on a floured surface.
In a small bowl, mix together the filling ingredients, then spread evenly on the dough, leaving an inch border on one of the short ends.
Starting at the other short end, roll the dough into a log shape and press down on the bare edge to seal.
With a sharp knife, cut into 8 equal slices.
Place slices on a parchment lined round cake pan, cover with a kitchen towel, and let rise for 1/2 hour in a warm spot.
Meanwhile, preheat oven to 350 degrees.
Bake the rolls for 20-25 minutes, or until slightly golden.
Let sit for 5 minutes.
Meanwhile, blend together icing ingredients in a blender, until completely smooth.
Spread the icing onto the warm rolls and serve.