Combine the milk, sugar, shortening, and salt, and heat until the sugar has dissolved and the shortening has melted. Let cool.
Dissolve the yeast in warm water and give it a few minutes to activate. Add the yeast mixture and the egg to the milk and mix well. Add the flour and mix to make a dough that is soft, but not sticky. Refrigerate overnight.
Remove the dough from the refrigerator and let it rise until doubled in size.
Preheat the oven to 300°F.
To prepare the filling for the cinnamon rolls, use an electric mixer to blend the brown sugar, cinnamon, and butter until softened and well mixed.
Roll out the dough, spread the filling on top, and tightly roll up the dough. Cut off the ends of the roll, cut the remaining roll into 1-inch pieces, and place the pieces in a greased 9x13-inch pan.