Spray the centers of four 20 by 12-inch sheets of aluminum foil with vegetable oil spray. Microwave olive oil, garlic, thyme, and red pepper flakes in a large bowl until garlic begins to brown (about 1 minute). Add potatoes, fennel, onion, sun-dried tomatoes, olives, and ½ teaspoon salt to the bowl. Toss to combine.
Pound the thicker end of each chicken breast between sheets of plastic wrap to uniform ½-inch thickness. Pat chicken dry with paper towels, then sprinkle with the remaining ¼ teaspoon salt and pepper.
Position 1 foil sheet with the long side parallel to the counter. Arrange one-quarter of the potato slices in two rows perpendicular to the edge of the foil. Top with one chicken breast and one-quarter of the vegetable mixture. Repeat with the remaining foil, potatoes, chicken, and vegetables. Drizzle any remaining oil mixture over the chicken.
Bring the short sides of the foil together and crimp tightly to seal. Crimp the open ends of the packets, leaving headroom inside. Refrigerate for at least 1 hour or up to 24 hours.
Adjust the oven rack to the lowest position and heat to 475°F. Place the packets on a rimmed baking sheet. Bake for 18–23 minutes, until the chicken reaches an internal temperature of 160°F. Let packets rest for 3 minutes.
Transfer packets to individual plates. Open carefully to release steam, then slide contents onto plates. Drizzle balsamic vinegar over the chicken and vegetables and sprinkle with fresh basil.