In the large bowl of a stand mixer, whisk together flour, yeast, sugar, and salt. Set aside.
In a medium microwave-safe bowl, combine milk and shortening. Warm in 5-second increments in the microwave (or on the stovetop), just until the mixture is lukewarm (about 110°F) and the shortening is starting to melt. If the liquid gets hotter than 110°F, just let it sit on the counter for a few minutes to cool. You want it to feel like warm bath water.
Pour the warm milk mixture into the dry ingredients. Use the paddle attachment on the stand mixer to mix on low speed (or medium-low) until combined.
Add the egg; mix until a soft dough comes together. Add more flour (1-2 tablespoons at a time), if necessary, until the dough pulls away from the sides of the bowl. It should be slightly sticky, but not sticking to the sides and bottom of the bowl.
Switch to the dough hook attachment on the stand mixer. Knead the dough for 5 minutes.
Transfer the dough to a large greased bowl; turn the dough so that it’s greased on all sides. Cover with a clean towel or plastic wrap. Let rise in a warm spot until doubled in size, about 2 hours.