Combine water, shortening, and 1/4 cup sugar; stir until shortening melts. Cool to lukewarm (105° to 115°).
Dissolve yeast and 1/2 teaspoon sugar in 2 tablespoons warm water; stir well. Let yeast mixture stand 5 minutes or until bubbly. Add yeast mixture to shortening mixture, stirring well. Stir in egg.
Combine flour and salt, stirring well. Gradually add flour mixture to yeast mixture, beating well. Cover; refrigerate at least 2 hours or overnight.
Punch dough down. Let rest 5 minutes. Heavily grease miniature muffin pans. Shape dough into 1-inch balls; place one ball in each muffin cup. Cover and let rise in a warm place (85°), free from drafts, 35 minutes or until doubled in bulk. Bake at 350° for 10 minutes or until golden brown.