Adjust rack to lower-middle position and preheat oven to 425 degrees.
For the filling, mix 3 Tablespoons soften butter, 1/2 cup packed brown sugar, 1 teaspoon cinnamon and all the nuts in a small bowl. Set aside.
On a floured work surface, roll the pie dough out and trim into a 12 inch square, then cut the dough into four 6-inch squares. Set an apple half on each square, core side up. Spoon a portion of the cinnamon mixture into each hallowed-out apple core.
Brush the perimeter of each square with egg white. Bring up the 4 corners of the pie dough, gently pulling to fit, around the apple. Pinch the edges and seal. Place on a large parchment-covered baking sheet. Brush dumpling tops and sides with egg whites. Add any pie dough embellishments (optional). Brush with egg whites. Mix 2 Tablespoons sugar with 1/8 teaspoon cinnamon. Sprinkle over the tops of dumpling. Repeat process with remaining apples and dough.
Bake dumplings until pastry sets and starts to turn golden brown, about 10 minutes. REDUCE heat to 375° and bake until golden brown, about 20 minutes longer.
While apples bake, bring 1 cup water, 2 Tablespoons soften butter, 1/2 cup brown sugar, and 1 teaspoon cinnamon to a boil in a small sauce pan over medium heat. Dissolve cornstarch and 2 teaspoons water; whisk into hot syrup for a smooth sauce. Cover, keep warm.
For each serving, spoon a portion of the sauce on a dessert plate. Set apple dumpling over warm sauce, serve immediately.