In a large mixing bowl, combine the warm whole milk and the instant yeast. Let it sit for about 5 minutes until it's frothy.
In a separate bowl, whisk together the all-purpose flour, sugar, and salt. Create a well in the center and add the vegetable shortening and egg.
Pour the yeast mixture into the dry ingredients. Mix until a dough starts to form. It will be slightly sticky but should come together nicely.
Transfer the dough onto a lightly floured surface. Knead the dough for about 5 to 7 minutes until it becomes smooth and elastic.
Place the kneaded dough in a greased bowl, cover it with a tea towel, and let it rise in a warm place for about 1 to 2 hours or until it doubles in size.
In a small bowl, mix together the sugar and cinnamon. This will be your sweet filling.
Once the dough has risen, punch it down to release the air. Roll it out into a rectangle about 1/4 inch thick. Spread a thin layer of vegetable shortening over the dough and sprinkle the cinnamon-sugar mixture evenly on top.
Starting from one long side, carefully roll the dough into a tight log. Use a sharp knife to slice the log into 12 equal pieces.
Place the rolls in a greased 9x13 inch baking pan, ensuring they are close but not touching. Cover them with a tea towel and let them rise for another 30 to 45 minutes.
Preheat your oven to 350°F (175°C). Bake the rolls for 20 to 25 minutes until golden brown.
While the rolls are baking, prepare the icing. In a medium bowl, whisk together the powdered sugar, vanilla extract, and enough whole milk to reach your desired consistency.
Once the rolls are finished baking, let them cool for a few minutes, then drizzle the icing generously over the top. Serve warm and enjoy!