Preheat oven to 425°F (220°C). Lightly grease a baking sheet.
Roll out each sheet of pastry to measure 12x12 inches. Cut into quarters, to make 8 - 6 inch squares.
In a small bowl, combine sugar, bread crumbs, cinnamon and nutmeg.
Brush a pastry square with beaten egg.
Place 1 tablespoon bread crumb mixture in center.
Place one apple half, core side down, over bread crumbs.
Top with another tablespoon of mixture.
Pull up four corners of pastry and pinch sides together to seal seams completely. Repeat with remaining apples.
Brush each dumpling with beaten egg.
Top with Pecan Streusel.
Place in preheated oven for 15 minutes, then reduce heat to 350°F (175°C) and continue baking 25 minutes more, until lightly browned.
Drizzle with icing.
Let cool completely at room temperature.
PECAN STREUSEL: In a small bowl, combine 2/3 cup chopped toasted pecans, 2/3 cup packed brown sugar, 2/3 cup all-purpose flour and 5 tablespoons melted butter. Mix with a fork until mixture resembles moist crumbs. Use as directed in recipe. Make more or less as desired.
To make icing, combine confectioners' sugar and vanilla and enough milk to make a drizzling consistency. Drizzle over cooled dumplings. Serve the same day.