Preheat oven to 425°F (220°C). Grease a baking sheet.
Roll out each pastry sheet to 12x12 inches.
Cut each sheet into 4 squares, making 8 total.
Mix sugar, breadcrumbs, cinnamon, and nutmeg in a bowl.
Brush pastry square with beaten egg.
Place 1 tablespoon breadcrumb mixture in center.
Place apple half, core side down, on breadcrumbs.
Top with another tablespoon of breadcrumb mixture.
Pinch four corners of pastry to seal completely.
Repeat with remaining apples and pastry squares.
Brush each dumpling with beaten egg.
Top each dumpling with pecan streusel.
Place in preheated oven for 15 minutes.
Reduce heat to 350°F (175°C) and bake for 25 minutes.
Dumplings should be lightly browned.
Drizzle with icing and let cool completely.