In a small saucepan heat whipping cream and butter until butter melts. Gradually add Parmesan cheese, stirring until combined. Cover and keep warm over low heat.
For a charcoal grill, grill chicken on the rack of an uncovered grill directly over medium coals for 12 to 15 minutes or until chicken is no longer pink (170 degrees F), turning once halfway through grilling. (For a gas grill, preheat grill. Reduce heat to medium. Place chicken on grill rack over heat. Cover and grill as above.)
Meanwhile, cook fettuccine according to package directions. Drain and keep warm.
Cut grilled chicken into bite-size pieces. In a medium bowl toss chicken with 1 tablespoon of the pesto. Add remaining pesto and Parmesan mixture to the hot cooked fettuccine. Add tomatoes. Toss to coat. Arrange fettuccine on a serving platter; sprinkle with pepper. Top with grilled chicken. If desired, garnish with pine nuts and/or basil. Makes 6 servings