Add warm water to a small bowl. Sprinkle yeast on top and stir to dissolve. Allow the yeast to activate (about 5 minutes).
In a large mixing bowl (or stand mixer), add the sugar, egg, salt, nutmeg, heavy cream, and shortening. Mix to combine.
Pour in the activated yeast and mix.
Add the flour to the mix. Use the dough hook attachment on a stand mixer and mix on medium speed until the ingredients are combined and the dough is smooth but still slightly sticky.
Form a ball with the dough and place it in a greased bowl. Cover and let rise for about 1 hour.
Use a deep-sided, heavy pot and add 3 inches of vegetable oil. Heat over medium-high heat until the oil reaches 370°F. Use a thermometer to monitor the temperature. Oil that is too cool can make the beignets greasy, while oil that is too hot can brown the outside before the centers cook through.
On a lightly floured surface, roll out the dough until it is ½ inch thick. Use a Mickey Cookie cutter to cut out Mickey-shaped pieces of dough.
Working in batches, fry the beignets until golden brown, about 1 minute on each side.
Use tongs to remove the beignets from the hot oil. Place on a paper towel-lined plate to drain. While the beignets are still warm, dust them completely with powdered sugar. Serve immediately.