For the vegetable pickles: Put the radishes and carrots in a medium nonreactive bowl. Combine the vinegar, sugar, salt and 4 1/2 cups water in a medium saucepan and bring to a boil. Pour the hot mixture over the vegetables and allow to cool. Cover the bowl and refrigerate the pickles up to 2 days.
For the chicken wings: Heat the oil in a medium saucepan over medium heat. Add half of the ginger and garlic and cook, stirring, until soft and fragrant, about 2 minutes. Add the vinegar, fish sauce and sugar and bring to a high simmer; adjust the heat and cook, whisking occasionally, until the bubbles become foamy and the mixture is thick enough to coat the back of a spoon, 20 to 25 minutes. Remove from the heat and stir in the reserved ginger and garlic and the pepper. Let the mixture cool to room temperature and then refrigerate until chilled, at least 1 hour (can be made up to 2 days ahead).
Position an oven rack in the top position of the oven and preheat to 425 degrees F. Grease a rimmed baking sheet and arrange the chicken wings in a single layer on it. Roast until the wings are golden and crispy, about 45 minutes, flipping the wings with a metal spatula about halfway through cooking.
Transfer the wings to a large bowl and toss with the chilled pepper sauce. Arrange the wings on a serving platter along with the pickled vegetables, lettuce, cilantro and lime wedges.