Using a stand mixer, cream together the vegetable shortening, sugar, and salt until light and fluffy. Add the egg and mix to incorporate. Add the flour, yeast, and milk and mix on low speed until a dough ball forms.

Switch to a dough hook and knead for a few minutes, or knead by hand for about 6 minutes, until the dough is silky, supple, and tacky, but not sticky.

Lightly oil a large bowl and transfer the dough to the bowl. Roll the dough around to coat it with oil. Cover the bowl with plastic wrap and let the dough rise until doubled in size, about 1 1/2 to 2 hours.)

Spray the counter with oil and transfer the dough to the counter. Gently roll out the dough into an approximately 15x9-inch rectangle. The dough should be about 2/3 of an inch thick. Do not roll the dough out too thin.

Combine the sugar and cinnamon in a small bowl and mix well. Sprinkle the cinnamon sugar over the dough evenly to the edge.

Gently roll the dough into a cigar-shaped log. Press the edge of the dough into the log to seal it. Cut the log into 12 even pieces.

Line a rimmed baking sheet with parchment or a silicone baking mat. Place the cinnamon rolls cut-side down onto the baking sheet. Proof the dough by letting it sit at room temperature until doubled in size.

Preheat the oven to 350°F.

Place the baking sheet on the middle rack of the oven and bake for 20 to 30 minutes, until the cinnamon rolls are golden brown. Remove from the oven and let cool on the pan for about 10 minutes before transferring to a wire rack.

While the rolls are cooling, make the icing by combining the powdered sugar, vanilla extract, and the whole milk. You may need to add a little more milk—the icing should be thin enough to make a thick paste that falls like a ribbon when drizzled from a spoon.

After the cinnamon rolls have cooled for about 10 to 15 minutes, drizzle the icing over the rolls.