For the pickled carrots and turnips: Mix the shredded carrots and turnips with the vinegar and sugar. Cover and let pickle for a minimum of 2 days.
For the chicken: Combine the soy sauce, oyster sauce, sesame oil, five-spice powder and salt in a bowl; mix to blend. Add the chicken, turning to coat. Cover and marinate in the refrigerator for 2 days.
Pass the marinated chicken through the flour, egg and cornstarch, and then deep-fry until cooked through, about 5 minutes.
For the sauce: Combine the chicken stock, soy sauce, sugar, oyster sauce, sesame oil, salt and garlic in a pot. Simmer for 15 minutes. Remove the garlic cloves. Dissolve the cornstarch in a little water and stir into the sauce. Simmer another 5 minutes, stirring frequently. Let cool.
To build the sandwich: Spread a bun, top and bottom halves, with some sauce. Top with a piece of chicken, some pickled carrots and turnips, and some lettuce.