When buying jicama, look for smooth, relatively unblemished roots with thin skin and juicy flesh. To peel the ginger, use the edge of a spoon.
In a blender, combine lime juice, lemon juice, oil, soy sauce, honey, mint, and ginger. Process dressing until smooth. Set aside.
Place chicken in a medium stockpot. Add enough water to cover by 1 inch. Bring to a boil. Lower heat, and simmer until cooked through, about 10 to 12 minutes. Transfer chicken to a medium bowl; allow to cool slightly. Shred chicken, and toss with all but 2 tablespoons of dressing. Set aside.
While the chicken is cooking, prepare an ice bath. Set aside. Bring a medium saucepan of salted water to a boil. Add snap peas, and cook for 30 seconds. Using a slotted spoon, transfer to ice bath to cool.
In a medium bowl, combine snap peas, arugula, and pea shoots. Toss with 2 tablespoons of dressing. Season with salt and pepper.
Add jicama and mango to chicken. Toss to combine. Season with salt and pepper. Cover each lavash with arugula mixture. Divide chicken salad between lavash. Roll up to enclose filling. Cut each lavash in half on the diagonal, and serve.