For pesto, in a blender container or food processor bowl combine basil, apricot halves, pecans, Parmesan cheese, olive oil, and garlic. Cover and blend or process with several on-off turns until a paste forms, stopping the machine several times and scraping the sides.
Rinse chicken; pat dry with paper towels. Place each chicken breast half between 2 pieces of plastic wrap. Working from the center to the edges, pound lightly with the flat side of a meat mallet to 1/8 inch thickness. Remove plastic wrap. Sprinkle chicken with salt. Spread pesto on each chicken breast. Fold in sides of each piece; roll up from short end into a spiral, pressing edges to seal. Secure with wooden toothpicks.
Place the chicken on the rack of an uncovered grill. Grill directly over medium coals for 18 to 20 minutes or until chicken is tender and no longer pink, turning once halfway through grilling time. Brush with preserves the last 5 minutes of grilling. Makes 4 servings.