Combine 3 tbsp olive oil, lemon juice, balsamic vinegar, Dijon mustard, dried parsley, dried oregano, onion powder, garlic powder, 1 tsp salt, 1/2 tsp paprika, 1/2 tsp pepper, and red pepper flakes in a medium bowl or large sealable bag; whisk to make the marinade.
Reserve 3 tablespoons of the marinade in a small bowl for the vegetables, then add the chicken to the remaining marinade and turn to coat; let the chicken marinate at room temperature 15–30 minutes (or refrigerate up to 2 hours and bring to room temperature 20 minutes before cooking).
Heat 2 tablespoons olive oil in a large 12-inch cast-iron skillet over medium-high heat until hot.
Add the chicken and cook 3–4 minutes until golden on the first side, flip, cover, reduce heat to medium, and cook another 4–6 minutes until cooked through (internal temperature 165°F / 74°C) depending on thickness; transfer to a plate and tent with foil to rest.
Return the empty skillet to high heat and add 1 tablespoon olive oil. Add the corn and cook, stirring occasionally, until charred in spots, about 3 minutes.
Reduce heat to medium-high. Add the zucchini, yellow squash, cherry tomatoes, minced garlic, the reserved 3 tablespoons marinade, and 1/4 teaspoon salt; cook 3–5 minutes until the zucchini is crisp-tender and tomatoes begin to soften.
Remove the skillet from heat and stir in fresh basil and optional mozzarella balls until combined and slightly warmed.
Slice or serve the rested chicken alongside the vegetables; taste and adjust seasoning with additional salt, pepper, or lemon juice if desired.