Cook first 4 ingredients in a medium saucepan over medium heat, stirring constantly, until shortening melts. Cool to 110°.
Stir together yeast and 1/4 cup warm water in a 2-cup glass measuring cup; let stand 5 minutes.
Beat yeast mixture, sugar mixture, egg, and 4 cups flour in a large mixing bowl at medium speed with an electric mixer until blended. Gradually stir in enough remaining flour to make a soft dough.
Turn dough out onto a lightly floured surface, and knead until smooth and elastic (about 5 minutes). Place in a well-greased bowl, turning to grease top.
Cover and let rise in a warm place (85°), free from drafts, 1 hour or until doubled in bulk. Punch dough down; cover and chill 8 hours.
Pinch off small pieces of dough (with well-greased hands), and shape into 1-inch balls. Arrange in 2 greased 9-inch square pans.
Cover and let rise in a warm place, free from drafts, 30 minutes or until doubled in bulk.
Bake at 400° for 20 minutes or until golden.
Note: To make ahead, bake rolls at 400º for 12 to 15 minutes or until just browned. Cool. Freeze up to 3 months. To reheat, bake rolls at 400º for 5 to 7 minutes or until golden