1. Set aside four pecan halves; chop remaining nuts. In a bowl, mix chopped nuts with 1/4 cup caramel sauce.
2. Peel apples; core each, creating a 1-inch-wide hollow from top to bottom. Place apples in a bowl and coat with lemon juice.
3. On a lightly floured board, roll puff pastry into a 12-inch square. Cut into four equal squares and center an apple upright on each. Push nut mixture equally into apple hollows.
4. Brush egg mixture over a 1/2-inch-wide border around each pastry; reserve remaining egg. To wrap each dumpling, with floured fingers bring two opposite pastry corners up over apple and pinch tips together. Repeat with remaining two corners. Firmly pinch all tips together; pinch each seam, then fold over about 1/8 inch along the seam and pinch again, leaving no holes. Space dumplings evenly on a buttered 12- by 15-inch baking sheet.
5. Bake in a 375° regular or convection oven for 20 minutes. Dip reserved pecan halves in egg mixture (save remaining egg for other uses or discard); press a nut half on top of each dumpling. Bake until pastry is deep golden, 10 to 15 minutes longer in a regular oven, about 5 minutes in a convection oven.
6. With a wide spatula, transfer dumplings to plates. Accompany with remaining caramel sauce and ice cream, if desired.