In a small microwave-safe bowl, heat jam in a microwave oven at full power (100%) until melted, 30 to 45 seconds. Stir in soy sauce, basil, and cayenne.
Rinse lemons and cut crosswise into 1/8-inch-thick slices. Discard ends and any seeds.
Rinse chicken and pat dry; cut into 1-inch pieces.
Thread 2 pieces of chicken onto an 8- to 10-inch-long skewer. Fold 1 lemon slice around 1 dried apricot and thread onto skewer through both sides of lemon slice. Repeat to fill 4 skewers equally. Set skewers on a rack in a foil-lined 10- by 13-inch broiler pan. Brush jam mixture generously all over chicken and fruit.
Broil about 4 inches from heat, turning once, until chicken is no longer pink in the center (cut to test), 8 to 10 minutes total. If desired, set pan over high heat, add any remaining jam mixture, and stir, scraping up browned bits, until boiling; pour into a small bowl and serve with meat. Add salt and pepper to taste.