For marinade, in a blender or food processor combine the ½ cup basil, 2 cloves garlic, the vinegar, the 1 tablespoon olive oil, the pepper and salt. Cover and blend or process until almost smooth. Place chicken breasts in a large resealable plastic bag set in a shallow dish. Pour marinade over chicken in bag; seal bag. Turn to coat chicken. Marinate in the refrigerator for 2 to 4 hours, turning bag occasionally.
In a large nonstick skillet cook shallot and 2 cloves garlic in the 1 teaspoon hot oil over medium heat for 3 to 5 minutes or until tender, stirring occasionally. Stir in artichoke hearts, chicken broth and tomatoes. Simmer for 2 minutes or until most of the liquid is evaporated. Stir in half of the goat cheese and 1 tablespoon of the snipped basil. Cover with foil and keep warm.
Drain chicken, discarding marinade. For a charcoal grill, place chicken on the grill rack directly over medium coals. Grill, uncovered, for 10 to 12 minutes or until chicken is no longer pink (170°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 directed.) Serve chicken with artichoke mixture. Sprinkle the remaining goat cheese and remaining 1 tablespoon snipped basil over the chicken.