Mix together the brown sugar, pumpkin pie spice and salt in a small bowl until well combined.
Melt the butter in a medium nonstick skillet over medium heat. Add the apples and cook until beginning to brown, about 5 minutes. Add the lemon juice and 3 tablespoons of the brown sugar mixture and continue to cook until the apples are soft and the sugar is caramelized, about 10 minutes. Let cool.
Beat the egg with 1 tablespoon water in a small bowl. Unfold 2 puff pastry sheets and cut a circle from each using a 7-inch bowl. Use the scraps to form the head, neck, wattle, feet and wing: Cut a piece vertically off one side of one of the scrap pieces, about 1 1/2 inches wide at the top and coming to a point at the bottom. Cut about 3 inches off the top of this strip. Pinch the pointed end and fold up and over to form the head and neck of the turkey. Next, cut 2 small triangles (about 1 1/2 inches) at the base for the feet. Cut 2 small triangles out of each foot to form 3 toes, then pinch these slightly to make claws. Cut one 2-inch-long narrow triangle with a slightly rounded base for the wattle. Cut a 4 1/2-inch triangle off the opposite corner of the scrap dough for the wing. Cut 2 small triangles from 2 of the edges and score each of these edges slightly with a few lines.
Transfer one of the pastry circles to a parchment-lined baking sheet. Use the egg wash to attach the head, neck, wattle and feet (the pieces should slightly overlap onto the pastry circle).
Unfold the remaining 2 sheets puff pastry and brush one side of one sheet with egg wash. Sprinkle the remaining brown sugar mixture in an even layer over the egg wash. Top with the second puff pastry sheet. Using the creases in the puff pastry as a guide, cut off a third of the pastry rectangle to make 2 rectangles, one large (two-thirds of the original rectangle) and one small (one-third of the original rectangle).
Cut 10 horizontal strips from the large rectangle, each about 3/4 inch wide, being careful not to cut completely through to the other side (these will be the feathers). Brush the right half of the pastry circle with egg wash and top with the feathers piece, curving it to follow along the right edge of the puff pastry round. Crimp with a fork to seal and twist each strip.
Cut the smaller rectangle into 10 horizontal strips. Twist and place around and between the larger twisted strips to fill out the feathers.
Add the apple mixture to the center of the puff pastry round. Brush the outside with egg wash and place the second round on top. Use a fork to seal and press down the edges of the rounds, sealing the pieces and the apples inside the round. Brush all over with egg wash (including the twists). Place the wing in the center of the round and brush with egg wash. Bake until very golden brown and puffed, 45 to 50 minutes. Let cool before serving.
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