1. In a large bowl, whisk together olive oil, balsamic glaze or vinegar, tomato paste (if using), honey, lemon juice, Dijon mustard (if using), soy sauce (if using), garlic, thyme/oregano, paprika, salt, and pepper until combined.
2. Pat chicken dry, season with salt and pepper, add to marinade and toss to coat. Cover and refrigerate for 2 to 4 hours, or at least 15 minutes.
3. Heat 2 tablespoons olive oil in a large skillet over medium heat. Add firm vegetables, season lightly with salt, and cook, stirring occasionally, for 8 to 12 minutes until lightly browned and crisp-tender. Remove vegetables and set aside.
4. For oven baking: Preheat oven to 425°F. Place chicken and marinade in baking dish or ovenproof skillet. Bake thighs 25-30 minutes or breasts 18-35 minutes until internal temperature is 165°F.
For stove-top skillet: Heat remaining 1 tablespoon olive oil in skillet. Sear chicken over medium-high heat 6-8 minutes per side until golden and releasing easily. Pour marinade or glaze over chicken, reduce heat, and cook until sauce thickens and chicken reaches 165°F.
For grilling: Preheat grill to medium-high. Grill breasts 5-7 minutes per side until internal temperature reaches 165°F.
5. If cooked separately, return vegetables to skillet with chicken for a minute to warm and combine flavors.
6. Garnish with fresh parsley if desired and serve.