Skin chicken. Place chicken pieces, bone side up, on the unheated rack of a broiler pan. Broil 4 to 5 inches from the heat for 15 to 20 minutes or until light brown.
Meanwhile, in a bowl stir together oil, mustard, lemon juice, lemon-pepper seasoning, oregano, and cayenne pepper. Brush mustard mixture on chicken. Turn chicken; brush with remaining mixture. Broil for 5 to 15 minutes more or until chicken is no longer pink (170 degrees F for breasts; 180 degrees F for thighs and drumsticks). Makes 6 servings.