DO-AHEAD TIP: If using bamboo skewers, soak them in water for 30 minutes before broiling.
Preheat oven broiler.
In a small microwave-safe bowl, nuke the apricot preserves on full power for 30 to 45 seconds or until liquefied. Stir in soy sauce, basil and cayenne pepper. Rinse lemons and cut crosswise into 1/8-inch slices; discard ends and any seeds.
Thread 3 pieces of chicken onto each skewer. Fold 1 lemon slice around 1 dried apricot and thread onto each skewer, through both sides of lemon slice. Repeat this process to fill skewers equally then salt and pepper to taste. Lightly coat the rack of a broiler pan with olive oil.
Place skewers on the rack and brush generously with the apricot preserves/soy sauce mixture; broil about 4 inches from heat source, turning once, until chicken is cooked through, 8 to 10 minutes total.
If desired, place broiler pan on stovetop over high heat; add any remaining apricot preserves/soy sauce mixture, bring to a boil and whisk up all of the browned bits from the bottom of the pan. Pour into a small bowl and serve with the skewers.