1. Heat oven to 400 degrees F. Unroll one piecrust onto a work surface. Roll out slightly with rolling pin to 12 inches. Fit into bottom and up side of a 9-inch deep-dish pie plate. Prick bottom of crust with a fork; refrigerate while assembling filling.
2. In a small bowl, blend 1/2 cup of the granulated sugar, the light-brown sugar, cornstarch, cinnamon, salt and cloves. Set aside.
3. Cut apples into 1/2-inch pieces. Melt butter in a large skillet over medium heat. Add apples, lemon juice and sugar mixture. Cook 10 minutes, stirring occasionally, until pan juices thicken. Cool on a rack for 25 minutes.
4. Pour cooled apple mixture into prepared crust-lined pie plate. Unroll second pie crust and roll out slightly to 12 inches. Cut 1-inch vent hole in center of dough. Transfer dough to pie, centering over filling. Crimp edge together and flute decoratively.
5. Brush pie with egg mixture; sprinkle with remaining 1 teaspoon granulated sugar. Bake at 400 degrees F for 20 minutes, then reduce temperature to 350 degrees F and bake an additional 25 minutes. Cover pie with foil if it browns too quickly. Cool at least 1 hour before slicing and serving.