Preheat the grill to high or the oven to 425°F (218°C).
Slice the chicken breasts to make 4 pieces and pound each to an even 4-ounce thickness.
In a large bowl combine the diced potatoes, sliced bell pepper, sliced onion, carrot matchsticks and minced garlic.
Pour the olive oil over the vegetables and add the dried basil, oregano, parsley, garlic powder, onion powder, thyme, red pepper flakes (if using), and salt and pepper; toss to coat evenly.
Lay out 4 large sheets of aluminum foil. Divide the vegetable mixture evenly among the sheets and place one chicken piece on top of the vegetables on each sheet.
Top each chicken piece with 1 tablespoon of unsalted butter, then seal each foil packet tightly.
Cook sealed packets on the grill for 15–25 minutes (about 20 minutes typical) or in the oven for 30–45 minutes (about 35 minutes typical), until the chicken registers 165°F (74°C) with a meat thermometer.
Carefully open the packets away from your face to release steam. Drizzle with a little olive oil and top with chopped parsley, a squeeze of lemon juice, and freshly grated Parmesan; adjust salt and pepper if needed before serving.
Stovetop option: After seasoning, cook chicken in a large skillet over medium-high heat in 2 tbsp oil, 2–4 minutes per side; add 1 tbsp butter after flipping and cook until 165°F. Remove and keep warm. Add more oil and 1 tbsp butter to the skillet and cook the vegetables until crisp-tender, then plate with the chicken and finish with Parmesan and lemon.