Simmer the apple cider in a small saucepan over medium low heat until reduced to 3/4 cup, about 20 minutes. Place in a jar with a tight fitting lid and chill. Add the next 7 ingredients, cover jar and shake well. Refrigerate until ready to serve.
Heat 2 tablespoons of the olive oil in a large skillet over medium high heat. Saute the red onion slices in hot oil until soft and caramelized. Remove and drain on paper towels. Core the apples and slice into even 1/4″ slices. Fry the apples in the hot oil for 2 minutes per side, or until softened and browned. Remove from the pan and drain on paper towels. Peel the sweet potatoes and slice into 1/4″ slices. Place in the pan and fry in hot oil for 2 minutes per side, or until browned and soft. Remove from pan. Sprinkle lightly with salt on both sides. Place on paper towels to drain.
Pound each chicken breast between two pieces of waxed paper to 1/4″ thickness. Cut each chicken breast into 3 equal pieces. Place the beaten egg in shallow bowl. Whisk in 2 tablespoons of reserved vinaigrette. Place the flour, cinnamon, nutmeg, and cayenne pepper in a second small shallow bowl. Whisk to combine. Dip each breast section in egg mixture, then dredge in flour mixture.
Place the other 1 tablespoon of olive oil in the pan. Add the chicken pieces and cook for 2 minutes per side, or until cooked through. Drain on paper towels.
Place the arugula in a bowl. Toss with enough of the vinaigrette to coat. Divide between four individual serving plates. Place a piece of chicken on the arugula on one side of the plate. Put a sweet potato slice, apple slice and some red onion on chicken, overlapping slightly and fanning as you stack. Repeat with remaining slices ending with chicken. Drizzle with vinaigrette. Sprinkle top of each serving with blue cheese, chopped Granny Smith and toasted pecans.