In a large nonreactive bowl, toss the chicken with the five-spice powder, then with the juice of 2 of the limes, the soy sauce, and ginger. Cover and marinate in the refrigerator for at least 30 minutes and up to a full day.
Combine the oils in a large skillet set over medium-high heat. There should be at least 1/₄ inch of oil in the pan. When the oil is hot, remove the chicken from the marinade, allowing the excess marinade to drip back into the bowl, and fry, in batches if necessary to not crowd the pan, turning the pieces frequently, until well browned and cooked through, 15 to 20 minutes.
Meanwhile, make the dipping sauce. Combine the oyster sauce, 1 tablespoon of the hot pepper sauce, and the juice of the third lime and stir to combine. Adjust the seasonings with more hot sauce, lime juice, and black pepper to taste.
Garnish with the scallions and serve with white or fried rice, with a drizzle of the sauce over each piece of chicken and the remaining sauce on the side.