Preheat oven to 375º.
In medium bowl, combine shortening and sugar. Mix well. in Separate bowl, measure warm milk and add yeast. Whisk well to fully dissolve.
Add milk/yeast to sugar mixture. Add remaining ingredients. Mix very well, being sure to remove all lumps. Dough will be quite soft.
Take a piece of plastic wrap and lay it out so it covers a 13 1/2" x 13 1/2" square. Sprinkle sugar on the wrap. Lay ball of dough on top of that. Then place another sheet of plastic wrap on top of the dough. Pat the dough down into a roughly squarish pancake. Lift the top wrap up and reposition as needed. Use a rolling pin to roll out the dough. Occasionally reposition the wrap. You may occasionally flip the whole thing over. Make sure that when you're done you've got about a 13 1/2" x 13 1/2" square dough. You may think it's way too gooey, but it will work.
Remove top piece of wrap. Combine filling ingredients. Spread evenly across dough's surface. You may want to leave about a 1" sugar-free edge because as you roll the dough, some of the sugar shifts to fill this edge. Use the bottom piece of wrap to lift the sugary edge of dough and start to roll it toward the sugarless edge forming a long cylinder. Cut off or trim up the irregular ends of the "log". Cut into 8 or 9 slices of similar size, about 1 1/2" wide. Using dental floss works well to cut the log.
Place rolls into a greased round glass pie pan.
Bake about 20 minutes, until tops are lightly brown.
Combine powdered sugar, vanilla, and milk and stir until lumps are dissolved to make glaze. The amount of milk you use will depend on how thick you want the glaze to be. Drizzle over warm rolls.
NOTE: You may make the rolls the night before and stop short of baking them. Wrap them up in their pan and leave them in the fridge for the night. Bake them the next morning.
Serves 8 or 9