If using a whole chicken, remove the giblets and pat the chicken dry. Place the chicken breast-side down on a cutting board and use kitchen scissors to cut along both sides of the backbone to remove it; then flip the chicken breast-side up and press firmly to flatten (spatchcock).
Place the prepared whole chicken or chicken pieces into a resealable plastic bag or shallow dish.
In a small bowl, whisk together the olive oil, balsamic vinegar, lemon zest and juice, chopped oregano, chopped thyme, minced garlic, and crushed red pepper flakes until combined.
Pour the marinade over the chicken and rub it all over the skin and into any crevices. Seal the bag or cover the dish and refrigerate for 1–2 hours or preferably overnight.
Preheat an outdoor grill or a large grill pan to medium-high heat.
Remove the chicken from the marinade and season generously with kosher salt and black pepper.
Grill the chicken skin-side down, covered, for 10–15 minutes until nicely charred. Flip and grill another 10–15 minutes, then flip once more and continue grilling until an instant-read thermometer inserted into the thickest part reads 160°F.
Transfer the chicken to a cutting board and let it rest for 10 minutes before carving.
Serve the chicken garnished with fresh basil and additional oregano if desired.