1. To prepare Rolls, in large bowl, combine 1-1/2 cups flour and yeast. In small saucepan, combine 3/4 cup apple sauce, 1/2 cup milk, 2 tablespoons granulated sugar, margarine and salt. Cook over medium heat, stirring frequently, until mixture reaches 120° to 130°F and margarine is almost melted (milk will appear curdled). Add to flour mixture along with egg. Beat with electric mixer on low speed 30 seconds, scraping bowl frequently. Beat on high speed 3 minutes. Stir in 2-1/4 cups flour until soft dough forms.
2. Turn out dough onto lightly floured surface; flatten slightly. Knead 3 to 5 minutes or until smooth and elastic, adding remaining 1/4 cup flour to prevent sticking if necessary. Shape dough into ball; place in large bowl sprayed with nonstick cooking spray. Turn dough over so that top is greased. Cover with towel; let rise in warm place about 1 hour or until doubled in bulk.
3. Spray two 8- or 9-inch round baking pans with nonstick cooking spray.
4. Punch down dough; turn out onto lightly floured surface. Cover with towel; let rest 10 minutes. Roll out dough into 12-inch square. Spread remaining 1/4 cup apple sauce over dough, to within 1/2 inch of edges. In small bowl, combine remaining 1/3 cup granulated sugar and cinnamon; sprinkle over apple sauce. Roll up dough jelly-roll style. Moisten edge with water; pinch to seal seam. Cut roll into 12 (1-inch) slices with sharp floured knife. Arrange 6 rolls 1/2 inch apart in each prepared pan. Cover with towel; let rise in warm place about 30 minutes or until nearly doubled in bulk.
5. Preheat oven to 375°F. Bake 20 to 25 minutes or until lightly browned. Cool on wire rack 5 minutes. Invert each pan onto serving plate.
6. To prepare Icing, in small bowl, combine powdered sugar, 1 tablespoon milk and vanilla until smooth. Drizzle over tops of rolls. Serve warm.