Place chicken in a large, resealable plastic bag. Pour the vinaigrette over it, seal, and marinate in the refrigerator for at least 4 hours or overnight if possible. I like to flip the bag a few times to distribute the marinade.
When ready to eat, prepare the grill. Gently mix the tomato, feta, basil, and a few grinds of fresh pepper in a medium bowl. Allow to sit at room temperature until the chicken is ready.
Remove the chicken from the marinade, discarding the remaining marinade, and cook the chicken over medium heat for 7-8 minutes per side, depending on thickness, or until the chicken is just cooked through. Remove from the heat and allow chicken to rest for 5 minutes. If you do not wish to discard the leftover marinade, be sure to simmer it for a few minutes on the stovetop before serving alongside the chicken.
Conversely, coat a large skillet with oil, and place over medium to medium-high heat. Sauté the chicken breast halves for 5 minutes each side, or until the meat is no longer pink in the center.
Arrange the chicken on a serving platter or individual plates. Spoon the tomato/feta mixture over top of each piece. Drizzle a little balsamic vinegar over the chicken and garnish with extra basil, if desired. Serve and enjoy!