Heat grill to medium-low. In a large bowl, combine the preserves, vinegar, ginger, cayenne, and ¾ teaspoon salt. Set aside ¼ cup of the mixture.
Add the chicken wings to the bowl and toss to coat.
Grill chicken, covered, turning occasionally, until cooked through, 25 to 30 minutes, basting with reserved mixture in the last 10 minutes of cooking.
Meanwhile, in a small bowl, whisk together the oil, pesto, and lemon juice.
Divide the lettuce among plates and drizzle with the pesto vinaigrette. Serve with the chicken wings.