For the dough, pulse-process flour, butter, salt and sugar in food processor and stream in eggs. Knead dough, wrap in plastic and chill for 1 hour.
To make the filling, heat a skillet over medium-high; add oil, 2 turns of the pan. Add chicken and brown and crumble then add garlic and spices, and stir to combine. Add currants and deglaze pan with Calvados; remove mixture to a bowl.
Return pan to heat and add butter to melt. Once it foams, add apples, celery and onions, and stir 5 minutes. Wilt in greens; add salt, pepper and nutmeg and nuts. Remove pan from heat and let cool. Add the cheese to the cooled greens and apples, and combine with the chicken.
Preheat oven to 375°F.
Roll out dough and cut into 6 squares, each 7 inches in diameter. Place filling in a triangle shape on half of each square, fold over the dough and press the edges closed. Arrange turnovers on a baking sheet lined with parchment and press the edges with a fork to seal. Brush turnovers with egg wash and bake to golden and hot through, around 30 minutes.