Preheat the oven to 400°F. Line a sheet pan with parchment paper or foil. Place the chicken in a zip top bag or bowl and drizzle on 3 Tablespoons of the balsamic glaze. Let sit for at least 10 minutes. Place vegetables in a separate zip top bag with the olive oil, 1 Tablespoon of balsamic glaze, salt and pepper.
Alternatively, arrange the chicken and vegetables onto the pan in an even layer and drizzle olive oil and the balsamic glaze over the top. Add fresh basil if desired. Sprinkle salt and pepper over the top.
Bake for 30 to 35 minutes, or until the chicken is no longer pink and the vegetables are fork tender. The chicken may get done faster than the vegetables, so check at 30 minutes and remove the chicken before the vegetables if needed.
Before serving, add extra balsamic glaze, salt and pepper if desired.