Pour boiling water over shortening and next 2 ingredients in bowl of a heavy-duty electric stand mixer, and stir until shortening melts and sugar and salt are completely dissolved. Let stand 10 minutes or until about 110°.
Meanwhile, combine yeast and warm water in a 1-cup liquid measuring cup; let stand 5 minutes.
Add yeast mixture and egg to shortening mixture; beat at low speed until combined. Gradually add flour, beating at low speed 2 to 3 minutes or until flour is blended and dough is soft and smooth.
Place dough in a lightly greased bowl, turning to grease top. Cover and chill 8 to 24 hours.
Turn dough out onto a lightly floured surface, and knead until smooth and elastic (about 1 minute). Gently shape dough into 60 (1-inch) balls; place 3 dough balls in each cup of 2 lightly greased 12-cup muffin pans. (You will fill only 20 cups.) Brush rolls with half of melted butter.
Cover pans with plastic wrap, and let rise in a warm place (85°), free from drafts, 45 minutes to 1 hour or until doubled in bulk.
Preheat oven to 400°. Bake rolls for 8 to 12 minutes or until golden brown. Brush with remaining melted butter. Serve immediately.